---
author: Adam's Axiom
date: '2026-10-15'
guest: ''
layout: post.njk
source: https://www.youtube.com/watch?v=ScFf69X7gYU
speaker: Adam's Axiom
tags:
- game-theory
- nash-equilibrium
- mechanism-design
- incentive-compatibility
- subgame-perfect-equilibrium
title: 博弈论基础与双咖啡馆定价
summary: 本文作为博弈论指南，通过双咖啡馆定价案例阐述纳什均衡与子博弈完美均衡，分析机制设计中的激励相容问题，并探讨模型假设与预测偏差。
insight: ''
draft: true
series: ''
category: philosophy-wisdom
area: society-thinking
project: []
people: []
companies_orgs: []
products_models: []
media_books: []
status: evergreen
---
### 博弈论基础与双咖啡馆定价
这是一份经济学博弈论的完整指南。我们将从基础讲起：什么是博弈，以及任何单方改变自身选择都无法获益的均衡。随后我们将进入更复杂的主题。谈判涉及协议以及协议带来的利益分享。拍卖理论则解释了规则如何塑造报价与支付。我们将通过详尽的示例和自测问题，一步步探讨每个概念。你将理解如何得出每个答案，以及为什么它是合理的。
Original English
This is a complete guide to game theory in economy. We will begin from the basics: what is a game and the equilibrium at which no one benefits change your choice independently. Then we let's move on to more complex topics. Talks relate to agreements and benefit sharing from agreements. Theory auctions explains how rules shape rates and payments. We step by step let's consider each idea with detailed examples and questions for self-checks. You you will understand how to get there to each answer and why does that make sense.
Original English
So what is theory? games? Game theory studies solutions where your best choice depends on the choice another's, and theirs—from yours. Let's see what this means for two coffee shops that determine prices for tomorrow. That's what The solution lies. You you run cafe A, and cafe B—through road. Today every evening the owner decides tomorrow's price for a cup of coffee, and there is only two options. Leave current price or lower it.
Original English
No owner sees another's choice before making a decision, and after opening the price remains fixed for the whole day. You have temptation to lower the price. To be cheaper a coffee shop—this is an excuse for coffee shop customers B to cross the road. You you will earn less on each cup, but if it comes new enough people, it can compensate for losses.
Original English
The difficulty is that coffee shop owner B now sitting in his institution and considers the same choice. It is quite possible that in they will have the same idea. Your competitor in an awkward way developed the ability think. And in this That's the problem. Your tomorrow's profit depends on your choice and from their , so you can't evaluate your choice separately. You must take into account that everyone can choose from you.
Original English
To compare results, we we will measure them in daily profit. Profit is money, received from sales, minus expenses for these sales and the work of the coffee shop. IN In this lesson I just I will provide profit figures. for our fictional coffee shop, because the main thing is to compare decision. Later in we will be on course expect profit, based on prices, quantities and expenses. Let's assume, what if you both save the current prices, each coffee shop will receive 100 dollars profit per day.
Original English
Now let's assume that you lower the price, and cafe B leaves yours. You become a cheaper coffee shop. Let's assume this attract enough new customers to increase your daily profit up to 150 dollars. Even despite the fact that you earn less on each cup. So far discount looks like quite profitable.
Original English
Now let's assume that you both lowered your prices. No more coffee shops is not cheaper than the other, therefore customers lose their reason move. You are not became cheaper than another institution. You both became cheaper than were yesterday. Each of you earn less on each cup, and if lower prices are not will provide enough extra sales, profit will fall. In our For example, let's say, what additional sales not enough, and each the cafe will receive 60 dollars.
Original English
Here is yours. first question. Or guarantees a reduction prices are what you earn more than 100 dollars, which would get if you Did both keep their prices? Find out how yours will change profit if cafe B will retain its price, and then how is he will change if the cafe B will also lower the price. Stop here if want to try this make.
Original English
If cafe B will keep the price, your profit will increase from 100 up to 150 dollars. 150 minus 100 is equal to 50. So this is profit of $50. If cafe B too will lower the price, your profit will fall from 100 up to 60 dollars. 60 minus 100 is equal to minus 40. So, that's a loss of 40 dollars compared to if both cafes kept the prices. So The answer is no. One and the same choice can make you 50 richer dollars or 40 poorer dollars depending on a decision you don't you control.
Original English
Please note that what we have not proven. We have not proven that price retention is the best solution. If Cafe B will lower the price, and you will keep yours, you you can get more worse result, and we have not yet considered such an option. We only showed that there is no possible benefit guaranteed. You are not can you rate this solution, looking only for the result, for which one, how are you hope, will let you out another owner. Therefore, you still need to make a decision.
Original English
That do it right, you need three information elements, which are the main parts of any games. First—who makes a choice. Persons who make decisions, are called players. IN in our case it is two cafe owners. Second—what can choose every player. Our owners can to maintain or reduce their prices. We are conscious excluded others options such as coffee improvement or moving to the street from less competitors. This can to be important in real business, but now we we single out one decision to understand him. Third—what is he trying to do? to reach everyone player. Economists they call it win. The win is number that reflects value of the result for the player. Here the winnings are daily profit, and we we assume that everyone the owner just wants maximize it.
Original English
Here is the second question. Are you planning? tomorrow's sales, and You are concerned about two things. . One—whether it will rain. Another—will it introduce discounts next door bakery. What about this? requires you to think about the choice another person who makes a decision? Pause before than I will answer.
Original English
It bakery. Rain, of course, can harm business, but he doesn't weighs the prices before how to decide whether start walking. But here the bakery owner can. Both factors are uncertainty that must be taken into account, but the bakery owner— player with a choice and own benefit. So his decision may to depend on what, in his opinion, You will do it. Exactly like this uncertainty game theory adds to economy.
Original English
So, you you know who chooses what everyone can choose owner and what each yearns. But you still you can't choose between price reduction and its preservation, because one is missing result. If the cafe "B" will lower the price, and you save yours, how bad is this can it end? That answer this, we need a way display all combinations of choices simultaneously. So let's fill in the result, which we missed.
Original English
Let's say you keep the price until Cafe B is lowering its prices. Now cafe "B" becomes cheaper, and the reason to cross the road appears already in the other side. Part your customers go to them. You, as before, earn a regular salary the amount from each cup, but you sell a lot less, whereas opening costs almost no institution are changing. In our For example, let's assume, what does this bring you 30 dollars of profit. AND cafe "B", finding itself in a situation that you hoped, gets 150.
Original English
We finally have all four results. But you you can still do it wrong comparison. If both will keep prices, you will get 100, and if both decrease —60. Price retention looks better so far you won't remember that you don't you can choose the price for cafe "B". To us need a way compare elections that you really you control. We we will place the results into the table that is called a matrix payments. This name sounds scary though object to this not deserves.
Original English
In lines elections listed cafe "A": save or lower. IN columns elections listed cafe "B": save or lower. Find intersection of line and column, so that see the result a combination of these two decisions. Each the cell contains two numbers. In this table The first number belongs to to the player in line, cafe "A", and the second - to the player in the column, cafe "B". This order —an agreement that must be determined, because The numbers themselves don't say that. they will say.
Original English
Let's consider these four cells. If "A" keeps price and "B" keeps it, in The cell says 100. and 100. If "A" keeps, and "B" reduces, it says there 30 and 150. "A" gets 30, and "B" gets 150. If "A" lowers the price, and "B" keeps, we we get a mirror image display: 150 and 30. And if both decrease
Original English
prices, everyone will get 60.
Original English
This is the whole game. Two choices for A, multiplied by two options for B, give four combinations. More options would mean more combinations. From this conditions, one thing follows rule. When you make a choice for a cafe Ah, compare. Cafe A's performance. When you make a choice for cafe B, compare indicators cafe B. Don't compare profit A from profit B and not name the larger one them better strategy A. And not can choose bank account B from the menu. Try it read the value in cell before we we will use the table. Cafe A lowers the price, and Cafe B keeps its price. In which we cell? How many does B earn? AND how much less is it, than B earns when both cafes keep prices?
Original English
Stop here and find it yourself row and column.
Original English
Choice A gives us the string "to reduce", and choice B gives us the column "to save. In this the cell shows 150, 30. B—is a player column, so B gets the second number, 30 dollars. When both keep prices, B receives 100. 100 minus 30 equals 70, therefore, B's profit is 70 dollars less. If you answered 150, you found the right one cell, but read the winnings another player. This minor mistake creates a lot trouble.
Original English
Let's also divide three things that often unite. Decrease prices are an action, available to one to the player. Decrease the prices of both cafes are the result of their joint actions. 60 dollars is a win one player per such a result. The action has no consequences. one fixed winnings. That which brings a decrease prices, depends on of what it is with collides. There is one more. the difference that we will need later. Strategy-this is the full plan of how to play Everyone here the owner approves one decision, therefore the choice "save" or ""reduce" is already complete plan. In the game with several solutions strategy can sound like this: "Get out to the market, then expand if demand is high, and to stay small, if he is weak." A separate action is only part of this plan.
Original English
And remember that "simultaneous" not means that both the owners decide in one second. One can write down the price of 8:00 and another at 9:00. If no one sees choosing another over how to make it, it's still a game with simultaneous choice. What they know under time of choice, more important for the coincidence of time on their watches.
Original English
Table also depends on the situation around her. Both owners know available options and numbers profit. Everyone cares only about your own daily profit. There is no one between them. agreements, and these winnings do not include future rewards or punishments. We let's review these assumptions later, but first we have understand that we assumed.
Original English
Let's let's try this method in another situation. It separate model with with their players and own indicators winnings. Service delivery chooses morning or evening collection, and the cafe, which buys from them goods, chooses small Is it a large order? This time the service delivery is a player line. So, first the number belongs to companies, and the second—cafe. Morning and evening they give eight, 12. Morning and the big one is given by 14, 18. Evening and morning give 10, nine. Evening and great give 16, 15. These numbers are pure profit in dollars for this particular agreements on delivery. Everyone the player gives preference for more net profit.
Original English
Here question. Cafe does great order, and the company takes him in the evening. What gets a cafe? Then, if only the company passes into the morning, What happens to her? profit?
Original English
Stop and watch, Who owns each? numeric. Evening and great they give 16. 15. The cafe is column player, so it gets 15. If the company is moving in the morning, while order remains large, we move on to morning and the big one, which gives 14. 18. Company now gets 14 instead of 16. 16 minus 14 is equal to two, therefore, its profit falls by 2 dollars. Cafe profit increases from 15 to 18, this an increase of three, but this not the comparison that asked to do company.
Original English
Now let's go back to comparison, which tempted you to beginning. 100 against 60. These two results changed the prices of both cafe. If cafe B lowers the price, your real comparison—it's 30 for saving prices vs. 60 for her decrease, because price retention is not returns the result in 100 dollars. Table prevented us answer false question, but she didn't tell me what to do, because you still don't know, which column will you choose? cafe B. So how to choose action when your the best move, It seems to depend. from the choice you make Can't you see? Sometimes your the best move is only seems dependent from the choice you make You don't see. Way to find out—to take each of the options cafe B in turn, record it and compare with him own options.
Original English
Therefore, let us first assume, that B retains its price. You are cafe A, you you can keep the price and earn 100 dollars or lower it and earn 150. 150 minus 100 is equal to 50, so the price reduction gives you $50 more. We move between two lines within one column, so choice B remains unchanged throughout of the whole comparison.
Original English
Now suppose that B lowers its price. You you can keep the price and earn 30 or lower it and earn 60. 60 minus 30 equals 30, hence the price reduction gives you $30 more. Again, choice B remains unchangeable. You prefer price reduction in both situations, so you not necessarily know, which column will B choose? We say that for A price reduction strictly dominates her retention. Strictly dominant strategy provides the player with higher gain than any alternative, regardless of actions other players.
Original English
Now your turn check another owner. If A holds the price, compare the two possible winnings B. Then repeat. comparison for case if A reduces price. Does B also have strictly dominant strategy? Point specific numbers for confirmation of your answers, instead of to rely on only for symmetry tables. Do it. pause.
Original English
If A holds price, B gets 100, if he keeps his, or 150 if it reduces. If A lowers the price, B gets 30 if keeps his, or 60, if it decreases. B gives preference for reduction prices in both cases, therefore both owners have the same strictly dominant strategy. If everyone acts accordingly incentive, both reduce prices and receive 60 each. This strange result, because both owners would prefer to receive 100 dollars, not 60.
Original English
We have reached this point, no assuming that someone of them is acting unwisely. We only provided to each owner control over one's own at a price that, unfortunately, is the only price that they actually control. It is important comparison not 60 vs. 100 that they could earn money together. When another cafe lowers the price, maintaining your own will return the result in 100 dollars. This gives only 30. Each owner can change one thing solution, not to beat both.
Original English
Such a structure called a dilemma prisoner. Every player has an incentive to act on harm to another, regardless of choice the last one. When both do that, they find themselves in worse situation, than if they had chosen cooperative alternative. Dilemma arises from incentives, not through the presence of a prison or particularly unpleasant person.
Original English
Let's let's translate this into prison language terms so that we reasoned logically, and not just chosen the largest number. It separate fictional model. Two suspects are interrogated separately. Everyone can remain silent. or confess. If both are silent, each gets 1 year. If one confesses, and the other is silent, the one, whoever confesses, receives zero years old, and the one who was silent—three. If both confess, everyone gets two years. Let's assume that everyone is only concerned prison term. What should do the first suspected of response to the silence of another? And what? to do in response to confession? Or confession dominates over silence? Do it. pause, remembering that these numbers reflect costs (punishment). In case of the silence of another, confession gives zero years instead of one. In case of confession another, confession gives
Original English
two years instead three. Confess better in both cases, because it is smaller the number of years is better. If both admit, everyone will receive two years, although if both were silent, each I would get one each. year. Incentives have the same structure as in the game with the cafe, but with in another direction advantages.
Original English
Let's go back. to the initial cafe tables and intentionally change one number. In this option A has extremely loyal regular customers. If A leaves the price, and B reduces, A earns 90, not 30. Everything else remains unchangeable. If B leaves the price, And all prefers the same reduction, because 150 is better for 100. But if B lowers the price, and now prefers that to leave, because 90 better than 60. A has more no dominant strategies.
Original English
Here is the second one option, again starting from the initial table, and not from 90. Let's assume that if A leaves the price, and B decreases, this gives 60, so same as decrease against decline. If In leaves the price, reduction still gives 150 instead of 100. So, there is never a decrease worse, and at least in in one case it better. We call this is weak domination. Draw in in certain cases allowed. Be worse in certain In some cases, no. But pay attention to the fact that which means a draw. If A is sure that B will lower the price, A will receive 60 in any in any case. Therefore, leaving the price is just as good in response to this belief. Weak the dominated action is not can be discarded as completely impossible, on unlike strictly dominated.
Original English
From now on, when I I remember the game from the cafe, I I mean again initial table. 100, 150, 30 and 60. Domination gives us a powerful short way, but in variant with loyalty is the best A's action depended on B, and many games are just like this for everyone. The best choice every person depends on others, and the choice of others is up to them. So how do we find choices that make sense together, when no one has dominant strategy ?
Original English
Let's answer. a new example of this , separately from the price tables. Our two they want to hold a cafe common street West. Every owner must choose Saturday or Sunday, no knowing the other's choice. If they choose different days, each the west will attract not enough people to to be worth the effort. If they choose the same day, both win.
Original English
Let's assume that if both will choose Saturday, A will get 80, and B will get 60. If both will choose Sunday, A will receive 60, and B will get 80. If they will choose different days, both will receive zero. It expected clean proceeds from the event, provided for example, and not calculated. Both owners prefer a successful event than lack of action, but A more I like Saturday, and Sunday. Predominant short way is not will choose the day for A. If B chooses Saturday, A gives preference for Saturday, because 80 better than zero. If B chooses Sunday, A prefers Sundays, because 60 is better than zero. The best choice A varies depending on from choice B.
Original English
The choice that brings the player highest profit for a certain choice of others or at a certain beliefs about their actions, is called best answer . This concept is conditional. . It responds to the question "best" about what", not " "the best of all." If there were two elections equal, both would be considered the best answers. We are not must invent preferences, which not in numbers.
Original English
Here first exercise. THERE ARE four possible pairs days. Find all couples in which each the owner already chooses best answer on the actions of another. Then check each pair in two steps. First change only choice A. Then change only choice B. Stop here and try this make.
Original English
Saturday for both pass verification. If only A moves to Sunday, A's profit falls from 80 to zero. If only B moves to Sunday, B's profit falls from 60 to zero. Sunday for both also pass verification. A lost would be from 60 to zero, having crossed alone , and B—from 80 to zero.
Original English
None inappropriate the steam does not pass verification. Suppose A chooses Saturday, and B—Sunday, so everyone gets zero. And could improve the result to 60, having passed alone on Sunday. B could improve result up to 60, having crossed alone on Saturday. Other mismatched pair not is being checked in the same way. If A chooses Sunday, and B—Saturday, A could move to Saturday and get 80, or B could would move to Sunday and get 80. We one is enough player with one a beneficial change to reject the pair. We are not we demand that both the players acted simultaneously. Two couples survive, so this the game has two equilibria Nash in the pure strategies.
Original English
Nash equilibrium is a set of strategies, in which no player cannot improve own profit, changing your strategy alone, provided that strategies of others remain unchangeable. Others in words, everyone player at the same time plays the best response to everyone's actions others. In such games with one choice is pure strategy means choosing one action from confidence. Subsequently we will consider random elections.
Original English
Let's see what this is. the check says about the initial game with prices. Both are lowering prices—this Nash equilibrium. If competitor lowers price, any the owner who will switch to it on its own price retention, will see the fall profit from 60 to 30. Both keep prices are not equilibrium because everyone the owner can change decision alone and to raise profits from 100 to 150. But if both keep prices, each owner receives more than if both are decreasing. So, equilibrium can to be worse for each player than another one available result. Balance does not mean that the result is fair, efficient, friendly or especially impressive. This means that are one-sided changes do not improve player's winnings for conditions that others strategies remain unchangeable. AND to customers who not at all in this one tables, can to like the lower ones prices.
Original English
Can each a certain pair of elections is not pass the test? Here second new game, separate from the cafe. Two players secretly choose an eagle or tails. If their the elections coincide, A wins $1 from B. If they are different, B wins $1 from A. So, A wants a match, and B wants differences.
Original English
If both choose Heads up, A won, but B wants to change the choice to tails. If A chooses heads, and B chooses tails , B won, but A wants change the selection to tails for matching. What happens with the other two pairs? Stop and check them out. If both choose tails, A reached a match, so B wants to change choice for an eagle, so results were different. If A chooses tails, and B The eagle chooses, but wants change selection to eagle for coincidence. Each of four pairs gives someone has a chance beneficial change. Not there is a Nash equilibrium in pure strategies. This is quite reasonable. respond. We are not must circle least inconvenient cell, hoping that no one will notice.
Original English
Returning to street event, test gives us two workers plans. But the owners still have to choose one, and search did not suggest equilibrium And, what day will B. A choose? prefers Saturday. B provides Sunday preference. AND choice of different days leaves both of them with nothing. Tradition, extreme booking deadline or messages can help them to coordinate actions. So, what will force anyone owner to be sure that the other will come the same day day? And when confidence enough? And when the incentives themselves have change?
Original English
Let's consider first confidence because in some games confidence is the main problem. Cafe owners agree that common application for orders can help them both. There is a nuance here. Addition works only when when both will spend the weekend on connecting your menu. Everyone could instead spend weekend on improving oneself institution and receive smaller but reliable benefit. Before to name anyone from owners non-contact, Let's look at what they are. take risks. This is another one. new model, and this Once you win, you get points. benefits, not dollars. The score measures, as far as the owner
Original English
appreciates the result, and higher numbers are better. Each owner chooses joint project or safe project, not seeing the choice of another. If both choose common, each gets eight points. If A chooses common, and B chooses safe, A gets zero, because I spent weekend on connection to application that is not up to why not connects. B gets four points from improved institution. Change elections for the opposite, and the winnings will change in accordance. If both will choose safe option, everyone will receive four. Therefore, safe option gives four, regardless from the actions of another owner.
Original English
Here question. If A I am sure that B will choose common option, or should choose A common or safe? If A I am sure that B will choose safe, what then should you choose A? Explain both. answers in numbers, and then tell me why one universal the answer would be incorrect.
Original English
Pause here. Against the common, A gets eight from common and four from safe, therefore joint is better. Against safe, A gets zero from common and four from safe, therefore Safe is better. Universal instruction to cooperate ignores risk to be left alone. Universal instruction to protect ignores the benefit of oneself, when another person joins. B faces the same in numbers, therefore "both are common", and "both are safe" are pure balances Nash.
Original English
From a situation where both chose common, any the owner who alone switches to safe, loses from eight to four. From the situation, where both chose safe, any the owner who alone switches to common, loses from four to zero. This structure often called "deer hunting." The name comes from the contrast between valuable hunting, which requires cooperation, and smaller prey, which can be obtained myself. Trust has importance, because attractive choice depends on whether another person too will do.
Original English
Compare this with our initial price reduction game. There was a decrease better even against the cafe that held its price. Here is a choice safe option worse than that who is participating in joint project. The result of cooperation is already in equilibrium in this game. We need confidence and coordination, so that to achieve this, not automatic elimination incentive to betray.
Original English
It also different from an incident on the street. Both owners are there wanted to choose that one the same day, but A preferred Saturday, and Sunday. Even being confident that they are coordinated, they they still couldn't to agree on that, at which equilibrium to stop. In the game with with the addition of such there is no disagreement. Both owners put together first place choice. The difficulty lies in in order to believe that another owner will really do it.
Original English
Sometimes for coordination even not needed message. Two customers agree on meeting, but forget to specify, near which entrance to market. Both prefer meet near northern or south entrance than cross. Noticeable watch above the north entrance can do it obvious place meetings. Such option for default, that attracts attention, is called focal point. The clock is not definitely does north entrance better. He helps. to every person guess what will guess another.
Original English
Now the second part of the previous question: what happens when the owners want the opposite? Cafe have a common area unloading, and both want get the most convenient time for delivery. Everyone can to insist on one's own slot or to yield to another. We will continue. use points benefits.
Original English
If both insist, deliveries are blocked each other, and each gets minus 10. If A insists, and B inferior, A gets six, and B—two. If A is inferior, and B insists, A receives two, and B—six. If both are inferior, they use less convenient slots, and everyone gets three. It chicken game (game of survival). Profitably to insist against the one who inferior, but it is profitable to give in to the one who insists. Worst result—it's mutual insistence. On unlike the game with application, clean equilibrium here provide for different election. Every player I would like to be the one who will get the desired slot.
Original English
Find clean ones equilibrium. Check options: both inferior, both insist and two uneven results. Don't assume that the most peaceful couple there must be stable. Stop here and check one-sided deviation. Option «both insist" does not pass, because every player can improve the result from minus 10 to two, having given in. The option "both" "are inferior" also not passes, because everyone can improve with three to six, insisting. Version "A insists, and B inferior" passes. AND would drop from six up to three, having given in, and B would drop from two to minus 10, insisting. Reverse result passes from the same reasons. So, there are two equilibrium, and each the owner gives preference to the one where he insists.
Original English
Now let's try another one version. Let's assume, appeared in the compartment enough space for both deliveries. We we change only one result. When both insist, every owner now gets seven points instead of minus 10. All other results remain unchangeable. Against the one who insists, persistence gives seven, while concession gives two. Against the one who gives way, persistence gives six, while concession gives three. Insistence now strictly dominates concession. Elimination collision costs changed the strategic problem.
Original English
But A often will not be sure that will do B. Let's assume, A believes that B is the same probability will join application or play safely. Is it worth it? joint decision risk? And in games on like "heads or tails"» predictability leads to you will take advantage. That to cope with both cases, we let's transform opportunities on probabilities. Calculate the expected winnings and find out, when randomization in itself is a reasonable strategy.
Original English
Let's start with the application. AND believes that B is the same will probably choose common option or safe. We can write it down belief as probabilities. Probability 1/2 means 50% chance. For full set opportunities that are not can happen simultaneously, probabilities should in total give one or 100%. If A chooses common option, A gets eight points half time and zero points half the time. Half past eight—four. Half of zero—zero. Add them, and together it gives in average four points. If A chooses safe option, A gets four points, regardless of B's actions. Under such an assumption two choices are certain balanced.
Original English
That, what we just did calculated, is called expected gain. You multiply each possible win on its probability, and then add results. Here quick question. The offer gives profit of $60 with probability 1/3 and a loss of $30 from with probability 2/3. The alternative gives zero as guaranteed result. Where is the highest? expected cash win?
Original English
Stop and note the minus sign for damage. 1/3 of 60—this is 20. 2/3 of negative 30—This is minus 20. Expected winnings is zero, therefore both options equivalent to expected money. The offer is still provides for 60 profit or loss 30. Zero is average. value, not the third possible payment. By many independent repetitions with the same probabilities average value goes to zero. This is not means that everyone was I don't care. The one who values avoidance risk, can give advantage guaranteed zero. To simulate such person, we would used winnings that reflect their preferences for risk, not silently assumed that dollars—this is their usefulness.
Original English
Expected winnings also explain another the problem we left open: predictability. Let's say you always You play "stone". You made life extremely simple for anyone who wants to play "paper". Transition to fixed cycle «rock, paper, scissors» does not solve the problem when they study this cycle. Sometimes a good plan—to make your next action consciously unpredictable. Let's assign "stone, paper, scissors" winnings: one for victory, minus one for defeat and zero for a draw. Winnings of two players always in the sum give zero. This is a game with zero sum. One player's win—this is definitely a loss another. Our game with there
Original English
was no price in the cafe a zero-sum game . Both owners could become or richer, or poorer together. If your opponent uses every action with probability 1/3, "stone" has expected gain 1/3*0 + 1/3*-1 + 1/3*1. This is zero. « Paper and scissors also have zero expectation. No action and no combination actions do not exceed this common average. If you too randomize the same, the same also true for your opponent. A combination of each player is the best response to actions another. This is equilibrium. Nash in mixed strategies.
混合策略为纯策略赋予概率。在这里，每种纯策略就只是一个行动。这并不意味着机械地轮流交替，因为那样只会成为一种新的固定模式。“抛硬币猜正反”能这样运作，仅仅是因为它只有两个行动且概率各为 1/2，但这种概率水平之所以行得通，是因为那些博弈是对称的。
Original English
Mixed strategy assigns pure probability strategies. Here each pure strategy is just one action. This is not means alternating mechanically, what would be just new template. "Eagle or "tail" works like this only with two actions and with probabilities 1/2, but probability levels worked because these games were symmetrical.
现在来看一个具有两个行动且总和为零的新博弈，在这里“五五开”是错误的。两位玩家各自独立选择“左”或“右”。当双方都选择“左”时，A 得到两分，B 失去两分。当双方都选择“右”时，A 得到一分，B 失去一分。当他们的选择不同时，A 失去一分，B 得到一分。A 希望出现一致，尤其是在左侧。B 则希望出现分歧。如果 A 总是选择左，B 就会选择右。如果 A 总是选择右，B 就会选择左。无论是寻找一致的人，还是寻找分歧的人，都总想改变自己的选择，因此不存在纯策略均衡。
Original English
Here new game with two actions and zero in the amount where "fifty" "for fifty"—this wrong. Two players regardless choose left or right. When both choose left, A gets two points, and B loses two. When both choose on the right, A gets one, and B loses one . When their elections differ, and loses one, and B gets one . And he wants a coincidence, especially on the left. B wants disagreement. If A always chooses on the left, B chooses right. If A always chooses on the right, B chooses left. Or the one who looking for a match, or the one, who is looking for disagreement, always wants to change the choice, so clean there is no balance.
设 P 为 A 选择“左”的概率，因此 A 以 1-P 的概率选择“右”。这里有一个稍微奇特的做法：我们通过观察 B 的激励机制来求解 A 的概率。如果 B 要同时使用这两种行动，他就必须在这两者之间无差异。否则，B 就会把我全部的概率都投向最好的那个选项。
Original English
Let P be the probability of what A will choose left, therefore, A chooses right from with probability 1-P. Here a somewhat strange move. We we find probability for A, looking at incentives B. If B is going to use both actions, he has to be indifferent between them. Otherwise B would direct all my probability of the best option.
如果 B 选择“左”，A 的收益是 2*P 减去 1 减 P 的值。第二部分是负 1 加 P，所以合在一起就是 3P 减 1。如果 B 选择“右”，A 的期望收益是负 P 加上 1 减 P 的值，这变成了 1 减 2P。因为 B 的收益总是与 A 的获胜相反，所以恰恰当这两个表达式相等时，B 才会感到无差异。因此，这个方程描述了 B 的选择，尽管其中的未知数是 A 的概率。
Original English
If B chooses left, A's gain is 2*P minus the value of 1 minus P. Second part —this is minus 1 plus P, so, together it goes 3P minus 1. If B chooses right, expected gain A is minus P plus value 1 minus P. This becomes 1 minus 2P. Since B's gain is always the opposite to win A, B indifferent just then, when these two expressions equal. So, this is the equation describes choice B, although the unknown in it is probability A.
3P 减 1 等于 1 减 2P。两边同时加上 2P，然后再在两边同时加上 1。5P 等于 2，因此 P 等于 2/5。A 应该在 40% 的情况下选择“左”，在 60% 的情况下选择“右”。我们来验证一下：面对 B 选择“左”的情况，A 得到 2*0.4 减 0.6，等于 0.2；面对 B 选择“右”的情况，A 得到负 0.4 加 0.6，同样是 0.2。B 无论偏向哪个选项都毫无所获。
Original English
3P minus 1 equals 1 minus 2P. Add 2P to both sides, and then add 1 to both parties. 5P is equal to 2, therefore P is equal to 2/5. AND should choose left in 40% of cases and on the right at 60%. Let's check this. Against the fact that B chooses left, A gets 2*0.4 minus 0.6, which is equal to 0.2. Against what B chooses on the right, A gets minus 0.4 plus 0.6, also 0.2. B wins nothing, giving preference any option.
求出 A 的策略并不能直接让我们得到 B 的策略，所以现在轮到你了。设 Q 为 B 选择“左”的概率。现在让我们使 A 在“左”和“右”之间保持无差异。求出 Q 并计算每位玩家的期望收益。停一下，在这里注意看你求的是谁的概率。
Original English
Finding strategy A does not give we have strategy B, so Now it's your turn. Let Q be the probability of what B will choose left. Now let's make A indifferent between left and right. Find Q and calculate the expected everyone's gain player. Stop. here and watch out for that, whose probability are you find.
如果 A 选择“左”，A 获得 2*Q 减去 1 减 Q 的值，也就是 2Q 减 1 加 Q，即 3Q 减 1。如果 A 选择“右”，A 获得负 Q 加上 1 减 Q 的量，等于 1 减 2Q。让它们相等：3Q 减 1 等于 1 减 2Q。5Q 等于 2，因此 Q 也等于 2/5。把 Q = 0.4 代入任意一个表达式中：3*0.4-1 等于 0.2。因此，A 的期望收益是 0.2，而 B 是负 0.2。
Original English
If A chooses left, A earns 2*Q minus the value of 1 minus Q. This is 2Q minus 1 plus Q, which is 3Q minus 1. If A chooses on the right, A earns minus Q plus magnitude 1 minus Q, which is equal to 1 minus 2Q. Equalize their. 3Q minus 1 is equal to 1 minus 2Q. 5Q is equal to 2, therefore Q is also equal to 2/5. Substitute Q = 0.4 into any expression. 3*0.4-1 is equal to 0.2. Therefore, expected gain A is 0.2, and B— minus 0.2.
这个博弈对 A 有微弱的优势，因为左侧的一致带来的收益高于右侧的一致，即使双方玩家的行为都很理性也是如此。为什么不是 50/50？如果 A 在一半的情况下选择“左”，那么 B 选择“左”平均会给 A 带来 2*1/2-1/2，即半个点；而 B 选择“右”会给 A 带来 -1/2 + 1/2，即零点。B 就会乐于每次都选择“右”，因为 B 支付的就是 A 所赢得的。40/60 的比例消除了这种脆弱性差异。
Original English
The game has slight advantage for A, since the coincidence on the left brings more than a coincidence on the right, even if both players act reasonably. Why not 50/50? If A chose left in half of the cases, left choice for B would give A in average 2*1/2-1/2, that is, half a point. Choice on the right for B gave would be A -1/2 + 1/2, i.e. zero . B with pleasure I would choose the right. every time, because B pays what A wins. Ratio 40/60 eliminates this vulnerability the difference.
这是一种极小化极大考量。A 选择一种使自己的最差期望结果尽可能大的策略，而 B 则选择使 A 的最佳期望结果尽可能小的策略。在这里，他们在 0.2 的水平上相遇。这种保护针对的是平均值，而不是单局比赛。但尽管如此，在具体的某一场游戏中仍有可能输掉一分；而且如果 B 可预测地选择“左”，A 就必须放弃混合策略，直接选择“左”，每次都拿到两分。在更复杂的博弈中，仅有简单的均衡行动是不够的。你还必须检查未进入混合的那些行动是否会带来更多收益，以及各项概率是否落在零到一的范围内且总和为一。
Original English
It minimax consideration. And chooses a strategy that makes his worst expected result as much as possible , and B chooses the one that makes the best expected result And minimal. Here they meet at the level of 0.2. This protection concerns average value, rather than a separate round. But still, maybe to lose a point in a specific game, and if B is predictable chooses left, A has to refuse mixtures, choose left and receive two points each time. IN more complex games simple equilibrium Action is not enough. You should also check whether the actions that did not enter the mix, did not bring more, and also or probabilities lie within the limits of zero to one and in The sum is given as one.
### 伯特兰价格竞争模型
到目前为止，我们博弈中的收益都是直接给出的。现实中的公司必须通过价格、产量和成本来赚取它们。如果两家公司向同一批买家销售同一种同品类的咖啡豆，价格竞争带来的结果会与产量竞争不同吗？
Original English
So far, payments in our games were given to us ready. Real companies have to earn them through prices, volumes and costs. If two companies sell the same coffee grains of the same kind to the buyers themselves, or will give a price competition other result than competition for production volume?
让我们借助一个新模型来回答这个问题。这里不再是我们之前收益表中的咖啡馆老板，那里的任何数字在这里都不适用。两家公司向完全相同的客户销售相同的咖啡，而且两家公司生产每包装的成本都是一样的。在我们即将构建的示例中，这些公司可以在一场博弈中定价 30 美元，在另一场博弈中定价 50 美元。区别在于每家公司选择什么，以及这些选择如何影响价格。
Original English
Let's answer the this question is for with the help of a new models. This is not cafe owners from our previous tables payments, and none of those there are no numbers here applies. Two companies sell the same coffee to one and the same to the same customers, and production of each the package costs both firms are the same. IN examples that we now let's build these firms can set price 30 dollars in one game and 50—in another. Difference is that each firm chooses and how are these elections affect the price.
我们从公司选择价格的模型开始。客户知道这两种价格，并会从较便宜的公司购买。如果价格相同，需求就会被平分。每家公司都能为整个市场提供服务。在每家公司中，边际成本都是每单位 30 美元，没有固定成本。它们同时定价，且随后不能更改。
Original English
Let's start with a model where firms choose prices. Customers know both prices and buy from a cheaper company. If the prices are the same, demand is shared equally. Every company can serve the whole market. In each fixed costs are 30 dollars per unit, fixed there are no costs. They set prices at the same time and not can change them subsequently.
假设两家都定下了 50 的价格。任何一家公司都可以定一个略低于 50 的价格，以此占领整个市场，并赚取比只在半个市场销售时更多的利润。因此，50/50 并不是均衡。只要双方确立的共同价格高于 30 且仍能吸引买家，这种降价的压力就会一直存在。如果一家公司定价比另一家低，但仍高于成本，那么定价较高的公司就有动力去降低这个有利可图的价格。如果价格高到没有人买，公司也可以降低价格来吸引买家，同时保持正的利润率。
Original English
Suppose, both established price 50. Any firm could appoint price just under 50, to capture the entire market and earn more than when selling on half the market. Therefore, 50/50 is not balance. Such the pressure of the decrease price is valid when both establish a common price above 30, which is all still attracts buyers. If one the company establishes a price lower than another, but still higher than cost, a more expensive company has incentive to reduce this profitable price. If prices are so high that no one buys, the company can lower your price, to attract buyers, keeping while positive margin.
在每单位 30 美元的价格下，没有哪家公司拥有利润率，因此每家公司获得的利润都为零。如果一家公司提高价格，它就会把自己的客户丢失给竞争对手。如果它把价格降到 30 以下，虽然能吸引客户，但在每单位上都会亏钱。根据我们的假设，两家公司都将价格定在边际成本水平就是伯特兰均衡。边际成本是额外再生产一单位产品的成本。
Original English
At a price of 30 dollars per unit no company has margins, so each gets zero profit. If the company increases the price, it loses his/hers customers for the benefit competitor. If she lowers the price below 30, that attracts customers, but loses money on each unit. According to our assumptions, installation of both firms at the same price level marginal costs are Bertrand equilibrium. Marginal costs are additional costs for production still one unit products.
这是一个令人印象深刻的结果，其中包含的假设起到了决定性作用。完全相同的商品、知情的客户、充足的产能以及轻松改变价格——这些不仅仅是装点门面的注脚。如果客户喜欢某家店的位置或产品口味，微小的价格差异可能不会迫使所有人都去另一家店。
Original English
It impressive result, and embedded in it assumption play back decisive role. Identical goods, informed customers, sufficient power and ease of price change— it's not easy decorative notes . If customers like location or product taste one store, a small difference in the price may not be forced all go to another.
你的第一个问题涉及关于产能的假设。两家公司都把价格定在 50，但每家公司在此价格下只能满足一半的市场需求。我们还能断言小幅降价允许一家公司占领整个市场吗？停下来，找出哪一步变成了谬误，不要试图去求解一个全新的模型。不，不能。降价的公司仍然只能满足半个市场，因此期望通过吸引所有客户来获得的收益是无法实现的。这并没有给我们一个新的均衡价格，它只是表明我们之前的证明不再有效。找出错误的假设是有用的，但这并不允许我们随意捏造替代的答案。现在让我们来改变这个决定，舍弃这种限制……
Original English
Your first the question concerns assumption about production capacity . Both companies set the price at 50, but everyone can to provide only half of the market price demand at this price . Can we still to claim that slight decrease prices allow one the company to seize the entire market? Stop and determine which step turns out to be false, without trying to solve absolutely new model. No, not. maybe. A company that reduces price, still can to provide only half of the market, so expected gain from involving everyone customers are unreachable. This does not give us a new equilibrium price. It indicates that our previous proof no longer valid. Find false assumption useful, but it doesn't give permission to invent response to replacement. Now let's change it. decision. Let's discard limitation
### 古诺模型与产量决定
两家公司的产能再次变得足以按意愿生产任意数量。然而这一次，每家公司都必须在市场价格确定之前先行决定产量。不妨设想提前对咖啡豆进行包装，此时若出现过剩产量，不付出成本就无法消除。在该模型中，每家公司在选择产量时，都将竞争对手的产量视为既定，但总供给决定了买方支付的价格。
Original English
capacities to both companies again could produce as much as they will want. However, this once each must to decide on quantity before will be installed market price. Imagine bean packaging in advance, when excessive production impossible to eliminate without costs. In the model Every company is a fool chooses the volume production, perceiving volume competitor as given, but cumulative the proposal defines the price paid buyers.
Original English
We we will use simple numbers. Price per unit is 90 minus total output volume, and it cannot fall below zero. Production each unit still costs 30. For this mathematical example volumes can be fractional numbers. Firms A and B simultaneously choose volumes production from scratch and above, without permanent costs and differences in products.
Original English
Let's try. 20 units from A and 20 from B. General the issue is 40, therefore the price is 90 minus 40, that is, 50 dollars. Each unit brings a margin of 50 minus 30, which is 20 . Company A receives $1,000 in income and spends 600 on production, so its profit is 400 dollars. Company B does the same. We reached prices above cost price, but everything we still have to check, Does any of the companies want change the volume production.
Original English
Therefore, let's call it volume company A QA, and the volume Company B QB. Until price is positive, price minus cost price units of production equals 90 minus QA minus QB minus 30. This is simplifies to 60 minus QA minus QB. Multiply this margin by QA and you will get profit A.
Original English
Let us fix the volume B and let's call it 60 minus QB letter D. To find profit A, subtract QA from D, and then multiply the result on QA. We can rewrite this as D in square, divided by 4, minus square differences between QA and half D. If open the brackets in the second version, we will get the first one. At fixed volume B part D in a square, divided by 4, cannot to change. Square part never can be negative, and we subtract it. So, the profit is highest when this the square is equal to zero, what happens, when QA equals D divide by 2.
Original English
Mathematical analysis not needed. It rewriting describes only the part where the pricing formula is positive. If total volume production reaches 90 or more, price becomes zero, and each unit brings a loss of 30 dollars. Production zero brings zero, therefore such volumes can never be the best. And if D is zero or is negative, any positive number causes losses, therefore the best choice A—zero. Let's substitute value D back. That find the best one answer A, subtract the volume of B from 60, then divide result by 2. If the result is negative, And chooses zero. Best answer B has the same shape with changed names.
Original English
These graphs show, how much each company wants to produce for any volume competitor. IN symmetrical both in equilibrium the volumes are equal one and the same number, Q. Therefore, Q equal to half from 60 minus Q. Multiply both sides in two. 2Q = 60-Q. Add Q to both parties. 3Q = 60. Divide by three, and Q is 20. We got the same one result, as well as before, now tested for every possible volume.
Original English
Here is yours. second question. Leave the same demand, but increase unit cost products for both firms from 30 to 45. In answer formula the number 60 becomes 90 minus 45 . How much does it produce? every company in symmetrical equilibrium and what will be price? Pause and go through these steps. 90 minus 45 is equal to 45. Equation equilibrium: Q is equal to half of 45 minus Q. Multiply by two, add Q, and you get 3Q is equal to 45. So, each firm produces 15. Total volume production is 30, and the price is 90 minus 30, i.e. 60. The margin of each firms—that's 60 minus 45, which is equal to 15. So, profit is 15 multiply by 15, or 225.
Original English
Now let's return initial cost in 30 dollars, and ours introductory puzzle solved. Price competition for our hard assumptions Bertrand gave a price of 30. Commitment to quantity of goods at the same time they gave a price of 50 . Firms have neither become smarter, nor less smart. In the first small model price reduction could capture the entire market. In the second—an increase production also lowered the price, which received it herself firm. We changed choice, so we changed it consequences.
Original English
Clients see it differently than companies. In the model number of buyers, willing to pay from 30 up to 50, do not buy, although they value grains higher than theirs cost. During these simplified assumptions such lost deals are at the cost of efficiency limited edition. The company's profit and total profit from trade—this different indicators.
Original English
Until that both companies made a choice, not seeing solutions alone one. But the company can first build up power and leave to a competitor possibility answer. When alone the choice becomes noticeable earlier than the other, the sequence of actions can change the result. To analyze this, we will unfold table into a tree decisions.
Original English
So, let's let's draw something like this tree with new situation and fresh in numbers. Nothing. transferred from the market grain or previous cafe tables. New cafe reviews possibility of discovery next to the existing one , and the owner the latter announces, that on any A newbie is waiting for a price war. A beginner could accept it warning literally and to stay away. Or he could to put better question. When am I already I will open it, will it be still fighting the best choice for the owner of the establishment?
Original English
First a newbie, which we will call " participant", chooses: to enter the market or to stay away. If he stays aside, he gets zero, and the operating company, that is, an old institution, gets 100. If he included, operating company sees this and chooses: to fight or to divide market. Market division gives each firm 40. The struggle gives the participant minus 20, and operating company—minus 10 . These are net profits in dollars for the same period, and I always I will call first participant's winnings. Both companies know these numbers give preference for larger money. Reputation and hidden costs yet not included in this tree.
Original English
In the tree every point where someone makes a decision, is called a node. The participant node is leaving first, and the entrance branch leads to the node of the current companies. Natural temptation—to start with from the very beginning and to ask if it looks The entrance is intimidating. Instead, the reverse induction begins from the last decision. If market entry took place, active the player compares minus 10 from the fight out of 40 from cooperation. Collaboration is better. IN this node 100 units from absence competitor already unavailable, as it were hot player I didn't mention them.
Original English
Now let's go back to that, who is entering? He expects cooperation, so the input gives him 40, while abstinence from input gives zero. He enters. Predicted path: login, and then cooperation. We first solved later decision and used it for evaluation of the previous one. Reluctance of the actor player to see no competitor made a fight profitable after that, how the entry occurred.
Original English
Let's try an option. Suppose the current the player has so big cost advantage, that the fight after entrance gives him 50 instead of minus 10. The one, who enters, still gets minus 20 from struggle, cooperation gives 40 each, and refrain from entering remains unchangeable. What now? implies the reverse induction? Stop. and solve the final decision before the first.
Original English
In the node of the current player fight gives 50 , and cooperation—40, therefore he chooses to fight. Waiting for this, he, who enters, compares minus 20 from the entrance with zero from retention. He refrains from entrance. Forecast changed, because changed incentive to perform threat, not because the threat was written in larger font. IN this changed game the strategy of the one who enters—to refrain. Incumbent's strategy player—to fight in in case of entry, even if on predicted path it never works. We turn this on answer, because the solution the one who enters, depends on what would happen after
Original English
another choice. Here a little bigger tree.
Original English
Provider first proposes cafe low or high fee for delivery. Cafe sees fee, and then accepts or rejects its. So, the strategy The cafe should determine, what will it do after each payment. Accept low and to reject a high—it one strategy. Accept both—another. Just to say "to accept" is not full plan if the fee affects choice.
Original English
Now let's say, the supplier records the fee is hidden, and the cafe should accept or reject it, no seeing her. Cafe can't anymore choose to "accept, if low" and "reject if "high", because he doesn't know, in what situation is located. On a tree we connect the nodes making decisions that the player cannot to distinguish, in information set. The player must choose the same effect for everyone them.
Original English
Hidden fee still important for results. Cafe just can't to depend on her in in their actions. Painting individual branches on The cafe doesn't give paper surveillance cameras.
Original English
Here is your second one. question. Two companies sign sealed envelopes with prices. A signs first, and B later, but B cannot read envelope A. Can B choose another price in response to actual price of A?
Original English
Stop and explain what is not enough. No. B not saw price A, so no maybe on her react. Setting the price later in time, strategically, the same, which is simultaneous. B can act on your assumptions about price A, but later the move does not give any advantages, if the signal is not reveals choice A.
Original English
So, the reverse induction clearly works in finite games with full information, where everyone the player sees previous actions before making a decision. WITH by covert actions we must take into account information sets and add often belief.
Original English
To us also needed check cases equality of results. If the firm receives 40 both from the struggle and from cooperation, both options will be best answer, and the input will depend from choice. Please specify the existence of equality, and don't choose silently convenient answer.
Original English
Initial threat failed because fight after entry was unprofitable. Or can the company change game before entering, so the threat became reliable? And does it give first move automatic advantage? We will check. both options, having done exactly obligation part of the solution.
Original English
It would be profitable for the company, so that the competitor believes in price war. But on our tree fight after entry gives the company minus 10, while cooperation—40. There is a threat reliable when it execution is best answer at the moment of action. Be dissatisfied situation not enough.
Original English
Ordinary equilibrium Nesha can miss such an empty threat. Let's take this pair. strategies: do not enter for a competitor and to fight in case of entry for the company. Provided plan to fight, Denial of entry gives competitor zero instead of minus 20. So, refrain from entering—this is the best respond. Because the competitor does not enter, the company never have to act. Change of plan to cooperation has not changed her income would be 100. None the player does not benefit from changes in strategy, therefore the couple passes the test Nash. But if the entrance took place, the plan forced would like to get the company minus 10 instead of 40.
Original English
The plan exists only because that no one knows him checks. In this tree full information each acceptance point solutions together with everything further development events form a separate game. This smaller game called a sub-game, and the whole game too is considered one of them. Perfect for subgames equilibrium—is a set of strategies, which gives the best answers in each underheat, even in those to which the path does not lead. Here is the reverse induction gives this result. Enter and cooperate if there will be an entrance.
Original English
One way change it—real obligation. Let's assume that to how the participant the market will decide, the current player can purchase equipment, which will lower it future operations expenses, and the participant the market sees this purchase. With this equipment fight after entering the market brings to the current player 50 units operational profit to cost consideration equipment. Distribution the market still gives 40, and still fighting brings to the participant minus 20. So, after market entry the fight is now best answer current player.
Original English
The participant compares minus 20 from the entrance to market with zero from detention, and he stays away. But the equipment is not free of charge. Let's assume it is costs 30, paid in advance. Now we need to distinguish three indicators: operating profit from the market, sunk costs that have already been paid, and total profit, which is the difference between first and second.
Original English
Without equipment is being logged in market, firm share, and total profit current player is 40. With equipment entrance to the market is holding back, and active player stores 100 units operational profit. Its total profit is 100 minus 30, that is 70. 70 is better than 40, so in this case the obligation is justified.
Original English
Here is yours. first question. Increase the value equipment up to 70, keeping it impact on the operating profit unchanged. Will it be active? player anyway buy it? Give answer to two individual questions. Or there is a threat convincing after purchases, and is there bargain purchase in advance?
Original English
Stop and Think about it. After shopping threat everything still remains convincing. 70 already paid independently from fighting active player or shares the market, so it doesn't change comparison 50 out of 40. However, before buying the comparison looks like otherwise. Purchase hinders market entry and leaves 100 minus 70, which is equal to 30. Cancellation of purchase leads to the entrance to market and division profit, which gives 40. 30 less than 40, so the current player is not must buy equipment. The obligation may to be too expensive, to choose it. Victory in less confrontational impressive when she costs more than just common market use.
Original English
Commitment also must be visible or reliably understandable for the player on whom it should have an impact. Secret equipment can change what active player would actually do, but it won't stop participant who has there is no reason to believe that something has changed.
Original English
Commitment also important when firms choose volumes production. Let's go back to the market beans from our lesson about competing firms with its initials in numbers. Price is 90 minus total volume release and it is not may fall below zero. Each unit costs 30 dollars in production. When firms A and B chose simultaneously, each produced 20 units and earned 400 dollars.
Original English
Now A takes over obligations regarding volume first. B is watching this and only then does he choose. AND cannot change his/her production volume subsequently. We already know best answer firm B. If the volume production A smaller for 60, subtract it from 60 and divide the result by two, so find the volume production B. If production volume A is 60 or more, B nothing produces.
Original English
Let's call it volume A through Q. B corresponds to 30 minus half of Q, so general issue equals 30 plus half Q. Price is 90 minus this the sum equals 60 minus half of Q. Let's subtract cost price of 30 dollars, and A gets margin of 30 minus half Q for each unit. Profit A—this is Q multiplied by this margin, i.e. 30Q minus half of Q squared.
Original English
We can rewrite it's like 450 minus 1/2 multiply by the square the difference between Q and 30. A square cannot be negative, therefore the highest profit, when this difference is zero, at Q, which is equal to 30. Then B produces 15. General the output is 45, and price—45, which leaves margin of $15 per unit. And earns 30* 15, or 450. B earns 15* 15, or 225.
Original English
This formula covered only volumes less than 60, therefore we should check the rest. At 60 or nothing more B produces, but own Issue A lowers the price to 30 or less. Its the margin then becomes zero or negative. 30—this best obligations in general. It's a game of volume. Stackelberg. AND earns more than while simultaneously choice, because he
Original English
chooses,
anticipating
reaction
B.
Commitment
makes
your
own
business.
Your
second
question.
Suppose,
A
undertakes
produce
20.
What
produces
B?
What
is
the
price?
AND
How
much
does
A
earn?Original English
Use
reaction
B,
not
consider
that
B
remains
at
the
level
15.
Let's
stop
here.
For
B
60-20
equals
40.
Dividing
40
by
2,
we
get
the
volume
production
20.
General
release
is
40,
the
price
is
50,
and
A's
margin
is
$20.
So,
A
earns
20*20,
that
is,
400,
the
same
amount,
How
many
companies?
were
elected
simultaneously.Original English
Obligation
produce
30
boosts
profit
A
up
to
450,
even
though
the
price
is
falling.
Bigger
number
of
units
from
with
a
smaller
margin
everything
one
might
be
better
option.
Is
it
always?
helps
to
walk
first?Original English
Let's
go
back.
to
the
game
"heads
or
tails"
",
where
A
wants
a
match,
and
B
wants
disagreement.
If
A
chooses
is
open,
and
B
sees
this
choice,
B
simply
chooses
the
other
side,
and
A
loses
every
time.
When
players
chose
simultaneously
and
accidentally,
expected
gain
A
was
equal
to
zero.
Here
revealing
one's
move
worsened
the
situation
A.
So,
the
advantage
the
first
move
is
conclusion
from
these
comparisons,
not
prizes
for
coming
earlier.Original English
We
considered
only
one
interaction
at
a
time.
Let's
go
back.
to
the
cafe
owners,
who
decide
whether
lower
prices.
Or
maybe
loss
future
profits
save
today's
price
justified?
That
to
answer
this,
we
we
have
to
provide
tomorrow
a
certain
gain.
Let's
define
win
for
tomorrow
at
the
game
where
we
first
encountered
this
problem.Original English
IN
original
game
with
cafe
owners
chose
prices
for
one
day.
Price
reduction
was
the
best
answer
for
each
owner,
regardless
of
actions
another.
So,
both
lowered
prices
and
both
found
themselves
in
the
worst
situation.
But
let's
assume
that
they
are
expecting
to
meet
again
tomorrow
and
the
day
after
tomorrow,
without
last
known
day.
Today's
choice
now
can
change
that
that
another
owner
will
do
it
later.Original English
Tomorrow
is
not
creates
collaboration
magically.
He
creates
another
one
consequence
that
follows
calculate.
We
we will use
initial
daily
profits.
If
both
owners
keep
their
prices,
each
earns
$100.
The
one
who
lowered
the
price
alone,
gets
150,
and
the
owner,
who
saved
it—30.
If
both
will
lower
prices,
everyone
will
earn
60.Original English
Now
imagine
proposed
strategy.
You
keep
your
price,
while
none
of
owners
never
lowered
his.
After
any
decrease
you
reduce
the
price
in
to
each
subsequent
one
round.
It
is called
the
strategy
of
"strict
trigger",
because
its
punishment
never
ends.
Let's
assume
that
everyone
the
owner
sees
clearly
the
price
of
the
other,
and
both
understand
the
strategy.
Collaboration
between
firms
can
harm
customers.
This
same
calculation
can
also
describe
the
cooperation
we
Congratulations,
for
example,
caring
for
the
common
resource.Original English
Next
to
us
need
a
way
to
determine
how
much
important
future.
Let
the
delta
indicates
how
much
next
round
more
important
comparatively
with
this.
This
can
to
display
patience
,
probability
continuation
of
the
relationship
or
both
together.
Delta
is
at
least
zero
and
less
than
one.
If
delta
is
equal
to
0.8,
then
100
dollars
in
next
round
cost
80
dollars,
when
the
owner
evaluates
them
today.
Through
two
rounds
the
same
100
multiply
by
0.8
twice,
which
gives
64.
Delta
has
stay
below
units,
since
100
dollars
every
day
forever
without
reduction
did
not
have
b
of
the
final
amount.Original English
Here
your
first
question.
Does
it
do
simple
things?
availability
future
Is
cooperation
profitable?
Set
the
delta
to
1/2.
Price
retention
forever
gives
100
now,
then
50
in
today's
equivalent,
then
25
and
so
on.
Let's
say
one
the
owner
lowers
the
price,
and
the
other
keeps
it.
The
one
who
lowered
gets
150
now.
After
that,
both
owners
are
reducing
prices
forever,
so
payments
from
punishments
cost
30,
then
15,
then
7.50
dollars
and
so
on.
Compare
the
two
amounts.Original English
Here's
a
hint.
Flow,
decreasing
twice
each
round,
equal
to
first
payment
plus
half
of
it
own
amount.
Stop
and
try
it.
Let's
call
it
amount
of
cooperation
V.
All
after
the
first
100—this
the
same
stream
again
,
only
reduced
twice.
So,
V
equals
100
plus
half
V.
Subtract
half
V
from
both
parties,
and
you
you
will
get
that
half
of
V
is
100
.
So,
V
is
equal
to
200.
Punishment
Flow
works
the
same
way.
Its
the
sum
is
30
plus
half
of
it
himself.
Therefore,
half
of
the
amount
is
30,
and
whole—60.
Add
150
immediate
profit,
and
the
reduction
costs
210.
The
future
exists,
but
decrease
anyway
wins
by
$10.
Tomorrow's
payback
is
real.
She
just
not
enough
yet
road.Original English
So
how
much
should
become
more
important
tomorrow?
Let's
go
back.
to
the
total
delta
and
let's
use
that
one
the
same
method.
If
the
owner
keeps
the
price
forever,
cost
is
100
now
plus
delta,
multiplied
by
that
the
same
flow
again.
So,
V
is
equal
to
100
plus
delta
multiply
by
V.
Subtract
delta
multiply
by
V
of
both
parties.
This
leaves
V,
multiplied
by
the
difference
1
minus
delta,
which
is
equal
to
100.
So,
cooperation
is
worth
100,
divided
by
the
difference
1
minus
delta.Original English
If
the
owner
lowers
the
price
once,
he
gets
150
now,
then
60
later
each
round
from
tomorrow
and
beyond.
This
future
flow
is
it
worth
a
delta
multiply
by
60,
divided
by
the
difference
1
minus
delta.
Please
note
that
we
compare.
These
are
complete
ways
that
include
another's
answer
owner,
not
just
100
today
versus
150.
Collaboration
is
at
least
the
same
profitable
when
the
share
100
to
1
minus
delta
is
at
least
150
plus
60
delta
by
1
minus
delta.Original English
Multiply
both
sides
by
difference
1
minus
delta.
Because
she
positive,
direction
comparison
is
not
is
changing.
Left
side
becomes
100.
Right
side
becomes
150
minus
150
delta
plus
60
delta.
Two
members
from
deltas
unite,
therefore
the
right
side
is
150
minus
90
delta.
We
need,
so
that
100
is
at
least
this
amount.
Add
90
delta
to
both
sides
and
subtract
100
and
you
you
will
see
that
90
delta
must
be
at
least
50.
Divide
by
90.
Delta
must
be
at
least
5/9,
what
constitutes
approximately
0.56.
Exactly
at
5/9
owner
never
mind,
keep
or
reduce
the
price.
Above
this
level
future
losses
prevail
today's
temptation.Original English
Now
let's
check
delta
at
0.8.
Collaboration
is
100,
divided
by
0.2,
which
is
equal
to
500.
For
we
will
start
the
decrease
from
60
multiply
by
0.8,
which
is
48.
Divided
by
0.2,
this
gives
240.
Add
150
immediately
profit,
and
decrease
costs
390.
Collaboration
wins
by
110.
We
also
must
check
punishment
from
that
the
very
reason
why
we
checked
threats.
When
both
the
owners
have
reduce
the
price,
or
Will
any
of
them
want
forgive
the
loan
and
maintain
its
price.
IN
in
case
of
reduction
by
others,
retention
gives
30,
and
the
decrease
gives
60.
There
is
still
a
decrease
the
best
answer,
therefore
the
punishment
is
quite
real.Original English
Support
for
cooperation,
however,
does
not
mean
that
it
must
happen
necessarily.
That
which
both
owners
lowering
prices
forever,
is
still
an
equilibrium.
Our
calculation
shows
that
cooperation
can
be
supported.
This
does
not
prove
that
the
owners
will
be
to
act
together.
What
about
as
everyone
knows
what
will
happen
exactly
three
rounds?
IN
last
round
price
reduction
is
not
will
lead
to
future
losses,
therefore
both
are
decreasing.
IN
second
round
result
of
the
third
already
defined,
therefore
both
again
are
reduced.
Same
The
logic
also
applies
to
first
round.
This
is
"
disentanglement"
depends
on
specific
assumptions
.
Single-round
game
has
a
unique
equilibrium,
and
the
final
date
known.
Winnings
are
not
contain
additional
social
motives.
Both
owners
they
know
all
this,
they
know,
that
the
other
knows
this,
and
and
so
on.
Change
these
assumptions,
and
conclusion
can
to
change.Original English
Here
is
yours.
second
question.
Suppose
the
owners
many
are
expecting
rounds,
but
their
monitoring
sometimes
records
a
price
decrease,
which
was
not
there.
Which
weakness
creates
"
"cruel
retribution"?
You
don't
need
equation.
Follow,
what
does
it
lead
to
false
report.
Stop
here.
One
false
report
dooms
both
owners
on
permanent
punishment.
This
strategy
strict
enough
to
to
restrain
intentionally
price
reduction
in
our
ideal
model.
She
is
also
quite
strict,
so
that
to
destroy
cooperationOriginal English
after a misunderstanding. If errors are possible, their probability and price must be taken into account in the analysis. Rule punishment does not become better only than that it is more dramatic.
Original English
"An eye for an eye" is a well-known alternative. You you start with cooperation, and then copy the previous one another player's action. It punishes for decline prices and allows return to cooperation. But after two players made mistakes, which use her, can to exchange blows in response. This is also not always there equilibrium or the best choice in each repeating game.
Original English
Reputation adds another level. Others people can perceive the action as information about What kind of player are you? Keeping the promise today maybe influence whether will be future partners to have with you matter. That simulate this, we must determine, who sees what, how are changing beliefs and what future opportunities it affects. Reputation cannot be mysterious additional winnings, which we only add when the initial The answer is inconvenient.
Original English
All this time we assumed that everyone the owner knows the winnings others. This is often not the case. . An existing business can to have low or high costs, and a new player can see the same price, not knowing with whom he has an opponent matter. Next we let's do these unknowns facts are part of the game.
Original English
Imagine that the new the player decides whether to open a cafe opposite already existing. He sees showcase and prices, but not sees expenses competitor. Acting business can have efficient kitchen and cheap products, or his expenses may to be so high, that aggressive competition to him will hurt. By the same facade can hide very well different incentives. Here the figures provided for this new model. Refusal from participation brings the new player gets zero. If he goes out on market vs. competitor with high costs, that divides the market, and participant earns $40. If he comes out against a competitor with low costs, he enters struggle, and a participant loses 20. These numbers already taken into account the reaction of competitors, and we will check them in the near future.
Original English
The competitor knows his own expenses, and new member—no. Let P be the probability that the costs high. Then the probability that costs are low, is equal to 1 minus P.
Original English
Pay attention to what there is not enough here. When the player cannot to watch by the actions of another player, as in the case of closed price, this incomplete information. When the player doesn't know something about the game itself, for example, about opponent's expenses, this incomplete information.
Original English
Our participant encounters another type. We describe privately known description as type. There is a competitor type with high expenses or type of low costs. AND because he knows, which one, his strategy can to determine that will do every type.
Original English
Expected winnings from the input is 40*P plus or minus 20*for the whole value 1-P. Multiply this second part. -20*1 is equal to -20. A -20*-P is equal to + 20P. So, we we have 40P-20 + 20P, which is 60P-20. Exit better than rejection, when this value is higher zero. Add 20 to both sides to 60P exceeded 20. Then divide by 60. Login better when P is greater 1/3. Exactly at 1/3 participant indifferent regarding the expected profit.
Original English
Your question. Member believes that there is 40% probability of high expenses and 60% probability of low expenses. Which expected income from entrance? How is this? compared to denied entry? AND what is the assumption about The risk lies in based on your recommendations?
Original English
Take a break. 40%of 40 is 16. 60% of -20 is equal to -12. 16-12 is equal to four. So, the input has expected profit 4 $ vs. zero for refusal. Member, who only cares about average money, enters the market. That, Who is especially afraid? lose $20, maybe to have other meanings utility, and then we compare the expected utility.
Original English
For next example we will change the initial belief. Previous participant's beliefs , before he saw any evidence, now is 30%high expenses. Imagine 100 comparable competitors. 30 have high costs, and 70— low. Certain feature contract supplier appears in 80% of companies with high costs and 20% with low. Let's assume that this the group describes our competitor, and participant just watching by this feature.
Original English
80% of 30—that's 24 companies with high costs, who have this feature. 20%from 70— these are 14 firms with low expenses that have its. This makes 38. firms with this characteristic, and 24 of them have high costs. So, if our active member has this feature, updated probability high costs is 24 out of 38. This simplifies to 12 to 19, or about 63%. It Bayes' rule in frequency form.
Original English
Pay attention to directionality question. 80%—it was probability of seeing sign provided high costs. To us was needed probability of high expenses provided presence of a sign. To confuse these things— a common mistake that does not give the correct answers.
Original English
Does it change? Is this a signal of a solution? Before observation: 60*0.3 is 18, and 18-20 is equal to -2. Therefore, new member stays away . After it: 60*12/19 is equal to 720/19. 20 is equal to 380/19. Subtraction gives 340/19, about $17.89. Signal transforms retention from entrance to entrance to market.
Original English
Now try the opposite observation, starting from that one the most a priori probability of 30%. The current member does not have of this sign. How many firms with high and low costs Don't they have it? What updated probability and would you enter market participant? Count the companies, not the change percentages intuitively. Do it. pause.
Original English
30-24 leaves six firms with high expenses. 70 minus 14 leaves 56 firms with low costs. These are 62 companies without a sign, therefore the probability high costs is 6 out of 62, approximately 10%. Now consider each of these 62 firms as equally likely opponent. Six of 40 of them are given to the participant, which in total is 240 . 56 of them give a minus 20, which in total is minus 1120. Together this is minus 880. Dividing by 62, we get approximately minus $14.19. The participant remains aside.
Original English
Absence of sign was also testimony. Us too need to check active participant, instead of to perceive it as weather. After login active participant with high costs earns 40 by dividing market, and -10, fighting, so he divides the market . Active participant with low costs earns 40 by dividing market, and 50, fighting, that's why he fights. These elections bring participant 40 and -20.
Original English
Now let's go back to initial a priori probability of 40%. According to this conviction, the participant enters market. High type divides the expenses market, and the type of low—fights. Exit is the best expected reaction participant in view of his beliefs. AND each type does their own thing own best choice. This is Bayesian Nash equilibrium. Everyone player and each type the player maximizes expected gain, considering your information and beliefs about others. In longer games belief change after previous steps, so we also need to check behavior after every possible history.
Original English
Our sign was passive. Now let's assume that the seller says that his product is great. Unscrupulous the seller can to say exactly the same thing . What could be done? this statement informative? Further we will compare an empty promise from guarantee. Then let's check if it's true in two types There are different sellers reasons for providing it.
Original English
Coffee shop needed new coffee machine. Two sellers are described their cars as reliable. This is useful, until we remember, that is dishonest the seller can also learn this word. If both types sellers will win from the trust of buyers and can do the same statement free of charge, by myself the statement gives the buyer little information. Such message called cheap chatter, because it sending to no one does not change the benefit because of expenses or obligation. Cheap chatter is not always useless. When two people just want find one place meetings, free the message works perfectly. The problem here is in another. Seller
Original English
low-quality the product wants to sound as well as seller of quality.
Original English
So, let's consider warranty. Seller promises to pay repair if the car will break. Quality cars break down less often, so one and the same the promise itself is worth it two types of sellers differently.
Original English
Here given numbers. If buyers believe that quality seller, they pay more, and seller's margin increases by 30 dollars for the car. Seller gets these 30. Also the seller pays warranty repair. IN average this the repair costs 10 dollars for quality machines and 50 for low-quality.
Original English
The buyer sees guarantee, knows these numbers and perceives it as a sign of high quality. Granting guarantees brings to a quality seller 30 units of benefit at 10 expected repair costs. 30 minus 10 is equal to profit of 20. So this the seller provides warranty.
Original English
If dishonest the seller imitates this, he gets the same 30, but pays 50. 30 minus 50 equals a loss of 20. So this seller is silent. During these assumptions only quality seller provides a guarantee, therefore buyers' assumptions are found correct. It separation signal.
Original English
Different types choose various observables actions, thanks to which the recipient will know something new. We checked both directions. Qualitative type must give signal advantage, and not silence. Low-quality type must give preference for silence, and not imitations. Show only that which is of good quality the seller can to afford guarantee, not enough.
Original English
Your question. Change expected cost repair low-quality seller from 50 to 20 dollars, leaving everything else unchanged. Will it continue? guarantee to share these two types? Calculate the benefit low-quality type from imitation.
Original English
Stop here. Now this guy gets 30 and spends 20, which gives 10 dollars of profit. By constant allowance in $30, $10 is better than nothing, because the low type will want copy. Both types will want offer warranty that violates proposed separation.
Original English
Cafe everything will get the same warranty service that has value, but the buyer who understands new costs, can't continue to count each guarantee confirmation quality. If he did that did, then would overpay 30 dollars for cars, which are often unreliable. Conviction, price allowance and choice the seller has to agree, and here this is not there.
Original English
In that example informed the seller chose an action. Could it be less? informed party to create the difference, offering a choice ? This approach is called screening.
Original English
Imagine delivery service, which doesn't know how much orders will be made client. Request to describe customers will encourage himself to optimistic answers.
Original English
Instead, the service offers a menu of two tariffs. For the same period light user is waiting two deliveries, and active—10. Base tariff does not have subscription fees, and the cost delivery is 8 dollars. Tariff « "membership" implies down payment 20 dollars, and then 4 dollars for each delivery. Service identical. Everyone the client knows their own usage volumes and chooses the cheaper one total amount. Clients only differ the number of necessary deliveries.
Original English
For the light basic user the fare costs two multiply by eight, that is 16 dollars. Membership costs 20 plus two multiplied by four, that is 20 plus eight or 28. Easy the user chooses basic rate. For active basic user The fare is 10 multiply by eight or 80. Membership costs 20 plus 10 multiply by four, that is, 20 plus 40 or 60. Active user elects membership. Their choice reveals the volumes orders without need to trust survey.
Original English
Your question. Reduce membership fee from $20 for four . What is the tariff now? will each customer choose? Does the choice allow membership still identify active user? Count both total amounts for membership. Do it.
Original English
Pause here. Base the tariff remains 16 for the lung user and 80 for active. Membership for the lung the user becomes four plus eight or 12, which is below 16. Membership for active the user becomes four plus 40 or 44, which is below 80. Both elect membership. Cheaper board can attract customers, but the menu doesn't have them anymore divides. So, exactly membership in itself says nothing about use.
Original English
Hidden quality before sale also can harm the entire market. Economists call this is unfavorable selection. Suppose, used car buyers technicians cannot distinguish reliable cars from unreliable, therefore offer a price, based on average quality. Owners of reliable machines know that their the goods cost more, so some refuse sell. Their output lowers the average quality of the machines that remain in sales. Buyers who they understand this, again are lowering the price, and this can replace the following the most reliable sellers. It possible mechanism, and not a statement that every second-hand market goods disappear. Inspections, guarantees and reputation can to change what they know buyers, and how are rewarded good sellers. Real market needs to be taken into account these features in models.
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Distinguish hidden quality and hidden efforts. If the car is already bad for sales, this hidden type. Now let's assume that equipment already paid, and he decides how much carefully to service the car . This is a hidden action, which is often called moral hazard, and we will develop incentive contract for this later.
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Let's assume that the cafe is all finds supplier to whom can trust. One the problem remains . The machine can be much more valuable for a cafe, how much does it cost give it to the seller. The deal can make both sides better, not determining how they will share this winnings. So, who is it? receives? Let's Let's define this in numbers.
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Let's say a cafe and this supplier can create 100 dollars total benefits by cooperating. Knowing that the deal creates value, not tells us how this the value will be distributed. For we need this to know that each the party can do, if negotiations will fail.
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Let's say a cafe can get 20 dollars from your next the best alternatives, and supplier—30. It their payments at differences that usually called external options. All values are specified in one currency units and are pure after taking into account relevant costs. 100 is the total amount, available from their agreements, not additional 100 over external options.
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Let's call the cafe's share in agreement X. Then the supplier receives 100 minus X. The cafe is not will agree to less than 20, because it can get 20 in another place. Provider will not agree to less than 30, which means that the cafe is not can get more 70. Therefore, acceptable range for cafes is from 20 to 70, and the rest goes to the supplier.
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How much extra creates value agreement? No deal external options in total give 50. With a deal total amount is 100. 100 minus 20 minus 30 leaves 50. So, the gain from The transaction is 50.
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The distribution under which each side gets half this additional I would give the cafe 20 if I won. plus 25, that is 45, and supplier 30 plus 25, that is, 55. Such equal division of winnings is one of the possible decisions, but he based on additional assumptions about as parties are bargaining. Knowledge external options gives us range acceptable agreements. It still doesn't give us the process, who chooses 45 and 55, not any other point in this range.
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Your question: increase external option cafes up to 40, while in supplier he remains at the level 30, and the deal is still on creates a general the sum of 100. What is it now? acceptable range for a cafe? As far as great benefits from agreement? And what will it give? equal distribution of these benefits? Hold total amount separately from additional profits.
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Stop here. Cafe now you need at least 40, and the supplier still need 30. So, cafe range—from 40 up to 70. The sum of external options is 70, and 100 minus 70 leaves 30 benefits from the deal. At equal distribution is 15 each. Cafe gets 40 plus 15, that is, 55, and supplier 30 plus 15 , that is 45. More strong alternative changed as acceptable range, and this particular one too distribution. Let's let's make
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rules clear in the new models. To isolate time, this game installs both payments in case of disagreement to zero. Therefore, previous 20 and 30 are no longer valid.
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First initiator offers a distribution of 100 dollars, and the other the party accepts or rejects. After deviation only 80 left dollars and roles are changing places. Primary defendant makes the final proposal. If it reject, both get zero. Proposals are being made in whole dollars, and the players agree, when they don't care.
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Start at the end. IN second round the defendant compares offer with zero from deviation. Zero proposal is equal to the deviation, and our rule is a draw says accept. Therefore, second initiator leaves himself 80 and offers zero.
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Now let's go back to first round. Primary defendant knows that refusal leads to save 80 later. These 80 are what what is it worth expectation. Therefore, the first initiator has offer at least 80.
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Proposition 80 leaves 20 to the initiator, while any smaller offer deviates and leaves the initiator with zero in the second round. 20 is better than zero, so the deal gives 20 to the first initiator and 80 to the defendant. First walk less impressively, when another person can to walk last. This the result depends from the rule final offer , reducing the amount and the rules are a draw.
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Here your second question. Leave your consent at indifference, but suppose that delay leaves only 60 dollars in second round instead of 80. What has to offer first initiator and how many he can leave itself? Work again in reverse order.
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Stop here. IN final round new initiator leaves himself 60 and offers zero, which accepted. Therefore, first defendant can provide 60, waiting, and immediate proposition 60 is acceptable. 100 minus 60 leaves 40 for the first initiator. More long delay weakens the one who answers, because worth the wait only that the player can get from smaller second fund.
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Eventually, distinguish transfers and losses. Transfer $10 from the cafe supplier changes distribution, but total amount one hundred remains. Reducing hundreds to eighty through delay destroys 20 dollars that no one receives. So the struggle for distribution can create losses that both sides would prefer to avoid.
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Now we can to say what creates agreement, with which each the party can leave and how time affects divide. But let's assume that several buyers they want the same car . Can change only payment rules place a bet equal value appropriate on one auction and disadvantageous on another?
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Before the cafe owner will make a bet, she should know how much will have to pay for the victory. Here it is. situation. The cafe wants a used car, and the seller is exhibiting her up for auction. IN different formats auctions wins the highest rate, but payment rule is different, and this the rule changes betting strategy.
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Let's start with the private values. Machine worth 80 dollars for this cafe because the benefit it provides can bring. And this one value is not changes when another participant evaluates car differently.
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Let's assume: one goods, closed bets , no commissions , budget constraints or plans resale. If the cafe wins, its surplus—this cost of the car minus what it is pays. At auction first price The winner is the one who proposed the most, and pays your bet. If the cafe puts 80 and wins, its surplus is equal to 80 minus 80, that is, zero. Rate 60 I would give 20 dollars. surplus in case victory, but inferior bet more often loses to others. Therefore, the owner balances between the chance of victory and size profit in case winnings. Only myself cost 80 dollars does not allow determine the best bid at auction first price. To us also needed assumptions about as, probably, will put rivals. Rate below the value that often called " understatement", defends excess. However rule, for example, always bet 10% less, does not have general justification for based on this information.
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On second price auction with closed betting winner pays the amount highest bid opponent. Your the bet decides whether you you win, and the bet decides the opponent, How much do you pay in in case of victory. By our assumptions about private value, bet on your the real value of 80 is weakly dominant strategy. It means that she never worse than any other bet, and sometimes better.
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Let's check this, recording the highest opponent's bet and comparing options.
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Let's assume the highest opponent's bet—60, which is below the cost for cafe. Bet 80 wins and costs 60, leaving 20 surplus. Bet 100 everyone wins too one costs 60, so adds nothing. Bet 50 loses and loses these 20. Underpricing It hurts when it interferes. win on price below your cost.
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Now let's assume that highest bid competitor—90, higher costs for the cafe. Bet 80 loses, what is good news. Cafe doesn't pay anything, but its excess is equal to zero. Rate 100 would win, but the cafe would pay 90 for that which is worth 80, so the surplus would be 80 minus 90, that is, minus 10 . Overpricing It hurts when it forces buy at price above your value.
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And if the rate competitor exactly 80, winning means payment of 80 for that is worth 80, while Lose—nothing. IN in any case the surplus is equal to zero. Below, above or equals. These cases cover all opportunities, and in none of them are different the bet doesn't win rate 80. This result regarding incentives. This is not promise that the cafe wins or not at all will receive any excess.
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Your turn. The cafe rates the car at 80 and bets 80 on second-price auction. Highest rate competitor may be 50, 75 or 90. Will it win? cafe in each of cases? How many Does it pay? And what surplus receives? Then explain how behaves the same bid 80 at auction first price.
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Stop here. Against 50 cafes will win, pays 50 and receives 80 minus 50, that is 30. Against 75 it wins, pays 75 and receives five. Against 90 it is loses, nothing pays and receives zero. At auction first price winning rate 80 means payment 80, therefore the surplus is equal to zero, no matter how low there was no bet competitor. Itself the rate has not changed. Payment rule changed the consequences rates.
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This result also depended on what the cafe knew own value. Therefore , let's change that for sale. Imagine, that the bidders want to buy a party coffee to resell its at market price. Resale value approximately the same for everyone, but no one doesn't know yet, and everyone has only an approximate assessment. It is called environment total cost.
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Suppose a party turned out to be worth 100 , and three participants had grades 80, 100 and 120. If higher grades lead to higher rates, most likely the winner has 120— most optimistic assessment. The winner is not received a certificate about higher knowledge. Maybe they just were the most optimistic person in the room.
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When bidders do not take this into account selection, victory often brings disappointment, and such the model is called " the winner's curse ". So, it's better questions not only in because, in my opinion opinion, it costs a lot, and in because it should be its value if my rate is the highest, because victory indicates that others rated it lower. This does not mean that winners always lose money or that there is a standard amount for deduction. Correct adjustment depends from how assessments are formed, how many people are taking participation in the auction and how they place bets.
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Here is a question about this difference. You You rate the picture at 80. dollars because you it's nice to hold her at home and you never her You will not sell. Another bidder values it at $100 . Does their higher mean estimate that your 80 dollars were wrong? Now compare this to the lot beans. Stop and decide what is private, what about common. Theirs pleasure not changes yours, so your private assessment can stay on levels 80. With beans evaluation of another the participant is information about that the most unknown resale value that you are trying to to determine. Therefore, their
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the rate can be evidence of your possible profit. That is why the proof for there is no second price here works. He predicted that the assessment The cafe remained at 80 unchanged as long as we changed the rate competitor. At shared value your rating is not known value, and Victory itself changes what you have expect. The goal in all this— excess, not victory for the sake of of victory itself.
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The buyer can win everyone auction, excessive overpaying. It effective way get both objects and pity. Economical the question is therefore, whether it exceeds expected benefit payment made.
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Pay attention to who selected payment method -seller. This one the choice changed the value of honest pond. Is it possible? consciously develop rules so that useful behavior became profitable for the one who accepts decision? This is a question most importantly, when people share the costs on what everyone wins.
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Our two cafes are just around the corner will face exactly with this kind of common expenses. Both of them want a better common delivery system. Every owner can pay for improvements, and every improvement brings benefits to both institutions. Owners agree that improvements are useful, but everyone would prefer, to have someone else pay.
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Now we can diagnose problem instead to read to them motivational speech about teamwork. Suppose each the contribution is worth payer 6 dollars and creates 4 dollars benefits for everyone cafe. These are conditional numbers, chosen in order to we could compare election. Benefits are added when both do contribution.
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If they pay both, every cafe gets four from own contribution and four from another, eight in total. Each the cafe also pays six, so each remains with the clean profit of two. If only A pays, then A gets 4 minus 6, that is, minus 2, and B gets 4, nothing paying. If only B pays, the numbers change in places. A gets 4, and B gets minus 2. If nobody pays, both receive zero.
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Let's take a look at the situation from A. If B contributes, then the contribution is given by A 2, then as absence contribution is given by A 4. If B does not contribute, then the contribution is A minus 2, while the absence contribution gives A zero. IN better anyway not to contribute, so it is purely dominant strategy, and B is faced with such by choice. Both owners refuse and receive zero each, although contribution I would bring both. two each. We rediscovered the structure of the dilemma prisoner in other conditions.
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Problem ticketless passenger consists of because a person can to benefit, without incurring the costs of its provision. Extreme the case is pure public good, which is non-competitive and inalienable. Non-competitiveness means that use of the good not by one person reduces accessibility for others. Inalienability means that it is impossible or hard not to to allow for good those who don't pay.
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Our shared system improvement works that's right for these two shops. Ordinary delivery should not to be like this because The van has limited space, and boxes one cafe can to displace others.
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Your question. What total benefit from a contribution of 6 dollars for both cafes and which one net profit the donor himself? Then explain why the contribution may be useful for a couple, but unattractive for owner who pays.
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Stop and compare these two points of view. One contribution creates 4 dollars for A and 4 for B, therefore, the total benefit is 8 dollars. Subtract the cost of 6 dollars, and a joint net profit will be two. But the donor personally receives only four of benefits, while paying all six, which leaves minus two. Part of the benefit gets someone to another, therefore individual comparison differs from group.
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So what? the rule could change it individual comparison? One of options—this mandatory joint agreement. Both pay six in that alone if both accept the package, and then improvement is being implemented. Now none of owners cannot to stay in situations when he pays himself, and no one cannot get benefit without payment. The choice becomes: either package that brings two each, or the absence of an agreement that brings zero to everyone. So, for our assumptions, both can agree.
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It design perspective mechanisms. You start with what you want result and you ask which ones the rules will force people to act in such a way that to support him, considering their information and incentives. Auctions, contribution rules and contracts can to be like that mechanisms. The mechanism still has take into account that people can really to do, will they take they participate and what actually knows designer.
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Joint the package changed whether the owners wanted pay. Now a cafe a rule is needed that will change the desire employee to do the work carefully. It hires equipment for service equipment. Thorough work increases probability successful repair with 40 %up to 80%, but it costs equipment 12 dollars efforts. Success brings cafe 100 dollars of profit, and failure—zero. Cafe can watch success, but can't directly check the effort.
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First a cafe offers a fixed payment 20, that paid regardless of whether whether the repair was successful. With careful work technician gets 20 minus 12, that is eight. Effortless technician gets 20. If the technician only appreciates this one income, fixed the payment is not motivating effort. Cafe Wish to get quality work in itself is not a reward for its implementation.
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Let's say a cafe pays base 20 plus $40 bonus in in case of successful repair. With effort expected bonus is 80% of 40, or 32. Add base 20 and subtract the cost effort 12, then expected income the technique is 40. Effortlessly expected bonus is 40% of 40, or 16. Add the base, and The income is 36. Careful work now more profitable four. Contract stimulates efforts in this model, that is the desired action brings at least the technique same expected income, as well as alternative.
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Your turn. Keeping that the basic pay itself, find the smallest bonus for success, which will make an effort at least the same attractive as their absence. Effort increase probability of success by 0.4 and cost 12. Stop and compare additional expected reward with these expenses.
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Let's call the bonus B. Efforts increase chance to get it on 0.4, therefore additional expected reward is 0.4*B. To us it is necessary that 0.4 B be at least 12. Divide by 12 by 0.4, and B should not be less than 30. Exactly at 30 technician indifferent between two options for action. Any bonus over 30 creates a clear incentive to effort, and our $40 bonus overcomes this threshold. It shows us what action will choose a technician after hiring. Will they sign? they contract in general—this is a separate audit.
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Suppose a technician can earn 35 in somewhere else. At $40 bonus effort gives 40, which better than 35. When minimum bonus 30 efforts yield 20 plus 24 minus 12, that is 32. So, the technician goes to another place. At fixed payment 20 even better effortless option gives only 20. If we ignored external options, we would develop a contract that no one will not sign.
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Check also a cafe. Provided making efforts for contract with $40 bonus expected cost The cafe is 80. His expected the payout is 20 plus 32, or 52. So, expected net the profit is 28. Let's assume that in the case refusal of repairs the benefit is zero. According to the specified cafe assumptions provides advantage of this contract. This is not proves that such the contract is the best of the possible. We are not checked everything basic options pay and bonuses, and also ignored risks and unaccounted for quality.
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We found acceptable option, which stimulates to effort and profitable to both sides. In both stories we changed the criteria choice, and then checked if anyone will refuse agreements. However, each of these checks was based on numbers, which we ourselves have determined . So what's happening? when the model itself incorrectly describes people? Let's check this out. question on the initial game with pricing. If both owners
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keep prices down, everyone receives 100 dollars daily profit. If one lowers the price, and the other—no, the one who lowered, gets 150. A the owner who kept price, gets 30. If both decrease, everyone gets 60.
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The decrease was more profitable regardless from the actions of a competitor, so we predicted, that both will reduce prices. Tomorrow both keep prices down. Or game theory has undergone defeat? Perhaps, our thoughts were erroneous, or source data.
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False forecast is a reason for research, not reason to defend table from reality. But research is not means fabrication any story, to save respond. This means checking that we wrote down what they want owners that they could choose and what they knew.
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Let's start with of what they want. In our tables it was assumed that each owner only interested profit for the current day. Let's assume instead, that everyone the owner promised to maintain the price, and violation of this promises perceived as loss of 60 units. Now the utility is daily profit minus 60 every time,