SBA Center Second Grade Economic Analysis Lecture 4 (Chapter 5) Part 2 PDF

SBA Center Second Grade Sheet (4) Economic analysis Lecture 4 (Chapter 5) Part 2 Joe & Sherif Tawfik 0100 312 1093  SBA JOE & Sherif Tawfik - 1...

SBA Center Second Grade Sheet (4) Economic analysis Lecture 4 (Chapter 5) Part 2 Joe & Sherif Tawfik 0100 312 1093  SBA JOE & Sherif Tawfik - 1 Multiple Equilibrium The Nash equilibrium is a useful solution concept because it exists for all games. - The possibility of multiple equilibria causes a problem for economists who would like to use game theory to make predictions, since it is unclear which of the Nash equilibria one should predict will happen. - The possibility of multiple equilibria is illustrated in yet another classic game, the Battle of the Sexes. ‫يبق فيه ر‬ Nash Equilibrium ‫اكت من‬ ‫ر‬ ‫يعن المشكلة هنا لما ي‬ ‫ي‬ Battle of the Sexes The game involves Two players: - A wife (A) and a husband (B) who are planning an evening out. - Both prefer to be together rather than apart (‫)منفصلين‬. - Conditional on being together, the wife would prefer to go to a Ballet performance‫ن‬and the husband to a Boxing match. The normal form for the game is given in Table 5.5, and the extensive form in Figure 5.3. Pure-strategy - To solve for the Nash Equilibrium, we will use the method of underlining payoffs for best responses. Table 5.6 presents the results from this method. A player’s best response is to play the same action as the other. Both payoffs are underlined in Two boxes: - The box in which both play Ballet and also in the box in which both play Boxing. Therefore, there are Two Pure- strategy Nash equilibrium: (1) Both play Ballet and (2) both play Boxing. - The problem of multiple equilibria is even worse than it first appears. Besides the two pure-strategy Nash equilibria, there is a mixed-strategy one. - Note: it is assumed that Nash comes with an odd number therefore, when It comes even numbers → there is Nash in mixed strategy.  SBA JOE & Sherif Tawfik - 2 Mixed Strategy Computing Mixed Strategies in the Battle of the Sexes The mixed-strategy Nash equilibrium in the Battle of the Sexes unlike in Matching Pennies, the equilibrium probabilities do not end up being equal (1/2) for each action Battle of the Sexes Assume: (W) → The probability the wife plays Ballet. (h) → The probability the husband plays Ballet. Because sum of probabilities must be = 1. Because sum of probabilities must be = 1. (1- W) → The probability of wife plays Boxing. (1 - h) → The probability the husband plays Boxing. - Our task then is to compute the equilibrium values of W and h. The difficulty now is that w and h may potentially be any one of a continuum (‫)سلسلةنمتصلة‬ of values between 0 and 1, so we cannot set up a payoff matrix and use our underlining method to find best responses. Instead, we will graph players’ Best-response function Function giving the payoff-maximizing choice for one player for each of a continuum of strategies of the other player.  SBA JOE & Sherif Tawfik - 3 Best-response functions - Let us start by computing the wife’s best-response function. Wife’s best-response function. Three possibilities: (1) she may strictly prefer to play Ballet. (2) she may strictly prefer to play Boxing. (3) she may be indifferent between Ballet and Boxing. 1- If she strictly prefers to play Ballet, her best response is w = 1 ‫ر‬ ‫ لل‬%100 ‫هتبق‬ Probability ‫ سعتها ال‬Ballet ‫ه لو بتفضل ال‬ ‫ي‬ ‫ف ي‬, Ballet ‫ لل‬Probability ‫ دي ال‬W ‫علشان‬ W = 1 ‫يعن ال‬ ‫ي‬ Ballet ‫ر‬ 0 ‫هبيق‬ (1-W) ‫ه‬ ‫الل‬ Boxing ‫لل‬ Probability ‫و ده معناه ان فالحالة دي ال‬ ‫ي‬ ‫ي ي‬ 2- If she Strictly prefers to play Boxing, her best response is W = 0. ‫فده معناه ان ال‬, (1-W) ‫ه‬ ‫ر‬ ‫الل ي‬ ‫ ي‬Boxing‫ر‬ ‫ لل‬%100 ‫هتبق‬ ‫ي‬ Probability ‫ سعتها ال‬Boxing ‫لو بتفضل ال‬ 0 ‫هتبق‬ ‫ي‬ Ballet (W) ‫لل‬ Probability 3- If she is indifferent about Ballet and Boxing, her best response is a tie between W = 1 and W = 0; (In fact, it is a tie among W = 0, W = 1, and all values of w between 0 and 1)  SBA JOE & Sherif Tawfik - 4 Let us start by computing the wife’s best-response function. Suppose the husband plays a mixed strategy of Ballet probability (h) and Boxing with probability (1 – h). ‫(ن‬Wife’s best response) ‫نفنبش نوفنالن‬,‫ن‬Boxing ‫اللنهونالن‬ ‫)ن ي‬1-h(‫نونالن‬,‫ن‬Ballet ‫اللنهونالن‬ ‫)ن ي‬h(‫ن‬:‫نعندهناحتمالين‬husband ‫يعنننالن‬ ‫ي‬ ‫ن‬.‫ ده‬payoff ‫ بتاعت ال‬probability ‫ و اضبة يف ال‬payoff ‫عننطريقننانناشوفنكلن‬ ‫ي‬ (Wife) Expected payoff from playing Ballet: h (The probability of husband plays Ballet) × 𝟐 (Her Payoff in Box 1) + 1 - h (The probability of husband plays Boxing) × 0 (her payoff in Box 2). (h) × 2 + (1 – h) × 0 = 2h (Wife) Expected payoff from playing Boxing: h (The probability of husband plays Ballet) × 𝟎 (Her Payoff in Box 3) + 1 - h (The probability of husband plays Boxing) × 1 (her payoff in Box 4). (h) × 0 + (1 – h)× 1 = 1 - h ‫ر‬ ‫بق‬ ‫( ت ي‬2h) ‫ه‬ ‫الل ي‬ ‫( ي‬Expected Payoff from playing Ballet) ‫ الزم ال‬Ballet ‫ تختار انها تلعب‬Wife ‫علشان ال‬ (1-h) ‫ه‬ ‫الل ي‬ ‫ ي‬Expected Payoff from playing Boxing ‫اعل من ال‬ ‫ي‬ 2h > 1 - h 3h > 1 h > 1/3 ‫ر‬ Boxing ‫ سعتها هتفضل تلعب‬h < 1/3 ‫ لكن لو ال‬, h > 1/3 ‫ لو ال‬Ballet ‫هتبق عاوزه تلعب‬ ‫يعن‬ ‫ي‬ ‫ي‬ ‫ ن‬Indifferent ‫ سعتها هتكون‬, h = 1/3 ‫ و لو ال‬, Therefore, Wife’s best response is when: 1- h < 1/3 So, W = 0 → Boxing ‫علشان هتفضل تلعب‬ 2- h > 1/3 So, W = 1 → Ballet ‫علشان هتفضل تلعب‬ 3- h = 1/3 → Indifferent  SBA JOE & Sherif Tawfik - 5 Now, Let us computing the Husband’s best-response function. Suppose the Wife plays a mixed strategy of Ballet probability (W) and Boxing with probability (1 – W). Husband’s best-response function. There are Three possibilities: 1- He may strictly prefer to play Ballet. 2- He may strictly prefer to play Boxing. 3- He may be indifferent between Ballet and Boxing. 1- If He strictly prefers to play Ballet, her best response is h = 1 ‫ر‬ ‫ لل‬%100 ‫هتبق‬ Probability ‫ سعتها ال‬Ballet ‫ف هو لو بيفضل ال‬, Ballet ‫ لل‬Probability ‫ دي ال‬h ‫علشان‬ ‫ي‬ h = 1 ‫يعن ال‬ ‫ي‬ Ballet ‫ر‬ 0 ‫هبيق‬ (1-H) ‫ه‬ ‫الل‬ Boxing ‫لل‬ Probability ‫و ده معناه ان فالحالة دي ال‬ ‫ي‬ ‫ي ي‬ 2- If He Strictly prefers to play Boxing, her best response is h = 0. ‫فده معناه ان ال‬, (1-H) ‫ه‬ ‫ر‬ ‫الل ي‬ ‫ ي‬Boxing‫ر‬ ‫ لل‬%100 ‫هتبق‬ ‫ي‬ Probability ‫ سعتها ال‬Boxing ‫لو بيفضل ال‬ 0 ‫هتبق‬ ‫ي‬ Ballet (H) ‫لل‬ Probability 3- If He is indifferent about Ballet and Boxing, her best response is a tie between h = 1 and h = 0; (In fact, it is a tie among h = 0, h= 1, and all values of w between 0 and 1) W W 1-W 1-W ‫(ن‬husband ’s best response) ‫نفنبشوقنالن‬,‫ن‬Boxing ‫اللنهونالن‬ ‫)ن ي‬1-W(‫نونالن‬,‫ن‬Ballet ‫اللنه نونالن‬ ‫)ن ي‬W(‫ن‬:‫احتمالي‬ ‫ن‬ ‫نعندها‬Wife ‫يعنننالن‬ ‫ي‬.‫ ده‬payoff ‫ بتاعت ال‬probability ‫ و اضبة يف ال‬payoff ‫عننطريقننانناشوفنكلن‬ ‫ي‬ ‫ن‬  SBA JOE & Sherif Tawfik - 6 (Husband) Expected payoff from playing Ballet: W (The probability the Wife plays Ballet) × 𝟏 (His Payoff in Box 1) + 1 - W (The probability Wife plays Boxing) × 0 (His Payoff in Box 2). (W) × 𝟏 + (1 – W) × 0 = 1W (Husband) Expected payoff from playing Boxing: W (The probability the Wife plays Ballet) × 𝟎 (His Payoff in Box 3) + 1 - W (The probability Wife plays Boxing) × 2 (His payoff in Box 4). (W)× 0 + (1 – W)× 2 = 2 - 2W (2- ‫الل هو‬ ‫( ي‬Expected Payoff from playing Boxing) ‫ الزم ال‬Boxing ‫ يخت رار انه يلعب‬Husband ‫علشان ال‬ (1W) ‫ه‬‫ ال يل ي‬Playing Ballet ‫ من‬Expected Payoff ‫اعل من ال‬ ‫بق ي‬ ‫ ي ي‬2W) 2-2W > 1W 2 > 3W 2/3 > W ‫ر‬ , Ballet ‫( سعتها هيفضل يلعب‬W > 2/3) ‫ لكن لو ال‬, (W < 2/3) ‫ لو ال‬Boxing ‫بق عاوز يلعب‬ ‫يعن هي ي‬ ‫ي‬ ‫ ن‬Indifferent ‫ سعتها هيكون‬, W = 2/3 ‫و لو ال‬ Therefore, Husband’s best response is when: 1- 2/3 > W So, h = 0 → Boxing ‫علشان هيفضل يلعب‬ 2- 2/3 < W So, h = 1 → Ballet ‫علشان هيفضل يلعب‬ 3- W = 2/3 → Indifferent So, now we have: - One Nash equilibrium in mixed strategy - Two Nash equilibrium in pure strategy Ballet, Ballet (1,1) Boxing, Boxing (0,0)  SBA JOE & Sherif Tawfik - 7 3 Nash Equilibrium ‫كده انا عندي‬ !! ‫انه فيهم‬ ‫اختار ي‬ One suggestion would be to select the outcome with the highest total payoffs for the two players. This rule would eliminate the mixed-strategy Nash equilibrium in favor of‫ن‬one of the two pure-strategy equilibria. In the Two pure-strategy equilibria,‫نن‬total payoffs, equal to 3, exceed the total expected payoff in the mixed- strategy equilibrium. A rule that selects the highest total payoff would not distinguish between the two‫ن‬pure-strategy equilibria. To select between these, one might follow T. Schelling’s suggestion‫نن‬and look for a focal point. ‫نقطةناالتصالن‬ Focal point Logical outcome on which to coordinate, based on information outside of the game. For example, the equilibrium in which both play Ballet might‫ن‬be a logical focal point if the couple had a history of deferring to the wife’s wishes on previous‫ن‬occasions. Focal point depends on: (History, Culture and Place)  SBA JOE & Sherif Tawfik - 8 Sequential Games In some games, the order of moves matters..‫ هيعمل ايه‬player 2 ‫ ميعرفش‬player 1 ‫لكن‬, ‫ عمل ايه‬Player 1 ‫ عارف‬Player 2 ‫الفكرة هنا ان‬ For example, in a bicycle race, the last racer has the advantage of knowing the time to beat. Sequential games differ from the simultaneous games we have considered so far in that a player that moves after another can learn information about the play of the game up to that point, including what actions other players have chosen. The player’s strategy can be a contingent plan, with the action played depending on what the other players do. The Sequential Battle of the Sexes Consider the Battle of the Sexes game analyzed previously with all the same actions and payoffs, but change the order of moves. Rather than the wife and husband making a simultaneous choice, the wife moves first, choosing Ballet or Boxing, the husband observes this choice, and then the husband makes his choice. - The wife’s Possible strategies have not changed: She can choose the simple actions Ballet or Boxing - The husband’s set of possible strategies has expanded. For each of the wife’s two actions, he can choose one of two actions, so he has four possible strategies. Note, In the following table: “Boxing | Ballet” should be read as “the husband goes to Boxing conditional on the wife’s going to Ballet.” The husband can choose: 1- Always go to Ballet. 2- Always go to Boxing. 3- He can also follow her. 4- He can do the opposite. Given that the husband has four pure strategies rather than just two, the normal form, given in Table 5.9, must now be expanded to have Eight boxes.  SBA JOE & Sherif Tawfik - 9 The Normal form in Sequential version is more complicated than the normal form in simultaneous version of the game in Table 5.5. By contrast, the extensive form in sequential Version ( in Figure 5.5) , is No more complicated than the extensive form for the simultaneous version of the game in Figure 5.3. Husband ‫عل اليمي لل‬ ‫الل ي‬ ‫ و ي‬Wife ‫عل الشمال لل‬ ‫الل ي‬ ‫األرقام ي‬  SBA JOE & Sherif Tawfik - 10 To solve for the Nash equilibrium, we will return to the normal form and use the method of underlining payoffs for best responses. columns ‫عل اليمي فوق ال‬ ‫ه ال ي‬ ‫ ي‬Wife ‫ و ال‬columns ‫عل الشمال فوق ال‬ ‫الل ي‬ ‫ هو ي‬Husband ‫ال‬ A: A → When A choose Ballet, B will get a payoff of (1) A → When A choose Boxing, B will get a payoff of (2) B: B-When B choose (Always Ballet) (A) will get a payoff of (2) B-When B choose (always follow her), (A) will get a payoff of (2) B-When B choose (Opposite A), (A) will get a payoff of (0) B-When B choose (Always Boxing), (A) will get a payoff of (1) There are Three pure-strategy Nash equilibrium: 1. Wife plays Ballet, husband plays “Ballet | Ballet, Ballet | Boxing.” 2. Wife plays Ballet, husband plays “Ballet | Ballet, Boxing | Boxing.” 3. Wife plays Boxing, husband plays “Boxing | Ballet, Boxing | Boxing.”  SBA JOE & Sherif Tawfik - 11 Subgame-Perfect Equilibrium ‫مثال للعبة الفرعية‬ ‫ي‬ ‫توازن‬ How do we formally eliminate Nash equilibrium that are unreasonable? Subgame-perfect equilibrium rules out empty threats Proper subgame ‫لعبة فرعية مناسبة‬ Part of the game tree including an initial decision not connected to another in an oval and everything branching out below it. A subgame is a part of the Extensive form beginning with a decision point and including everything that branches out below it. A subgame is said to be proper if its top most decision point is not connected to another in the same oval. Figure 5.6 shows the extensive forms from the simultaneous and sequential versions of the Battle of the Sexes, with dotted lines drawn around the proper subgames in each. In the Simultaneous Battle of the Sexes, there is only one decision point that is not connected to another in an oval, the initial one. Therefore, there is only one proper subgame, the game itself. ‫نبيلعبوان‬2 Players (A&B) ‫نالن‬simultaneous ‫الفكرةنفنالن‬ ‫ي‬ ‫نفنعاندين‬,‫معنبعضنوكلنواحدنميعرفشنالتاننهيعملنايهن‬ ‫ي‬ ‫نن‬.‫نواحدهنبس‬Proper Subgame In the Sequential Battle of the Sexes, there are three proper subgames: the game itself, and two lower subgames starting with decision points where the husband gets to move. ‫نن‬Player (A) ‫ نبيلعبنبعدن‬Player (B) , Sequential‫يفنالن‬ Player ‫ناألولن وبعدينن‬Decision ‫نبياخدن‬Player (A) ‫يعنن‬ ‫ي‬ ‫ن‬Player‫نبتاعن‬Decision‫نكردنفعلنللن‬Decision‫(نبياخدن‬B) ‫(ن‬3 Proper Subgame) ‫نفنبقنعاندين‬, ‫ي‬ ‫(ن‬A)  SBA JOE & Sherif Tawfik - 12 A subgame-perfect equilibrium ‫مثال للعبة الفرعية‬ ‫ي‬ ‫توازن‬ Is a set of strategies, one for each player, that form a Nash equilibrium on every proper subgame. - A subgame-perfect equilibrium is always a Nash equilibrium. In the sequential version of the Battle of the Sexes, the concept of subgame-perfect equilibrium is important. In addition to constituting a Nash equilibrium on the whole game, strategies must constitute Nash equilibrium on the two other proper subgames. 1- In the left-hand subgame, following his wife’s choosing Ballet, the husband has a simple decision between Ballet, which earns him a payoff of 1, and Boxing, which earns him a payoff of 0. The Nash equilibrium in this subgame is for the husband to choose Ballet. 2- In the right-hand subgame, following his wife’s choosing Boxing, he has a simple decision between Ballet, which earns him 0, and Boxing, which earns him 2. The Nash equilibrium in this subgame is for him to choose Boxing. “Ballet | Ballet, Boxing | Boxing.” → Subgame perfect equilibrium. Returning to the 3 Nash equilibria, only the second one is subgame-perfect. The first and the third are not. For example, the third equilibrium, in which the husband always goes to Boxing, is ruled out as a subgame- perfect equilibrium because the husband would not go to Boxing if the wife went to Ballet; he would go to Ballet as well. Subgame-perfect equilibrium thus rules out the empty threat of always going to Boxing that we were uncomfortable with in the previous section. More generally, subgame-perfect equilibrium rules out any sort of empty threat in any sequential game. Note: Subgame-perfect equilibrium does not reduce the number of Nash equilibria in a simultaneous game because a simultaneous game has no proper subgames other than the game itself.  SBA JOE & Sherif Tawfik - 13 Backward Induction A shortcut ‫ن اختصار‬to find the subgame-perfect equilibrium directly is to use backward induction. - A simple way to find the subgame perfect equilibrium. Backward induction works as follows: (1) Identify all of the subgames at the bottom of the extensive form (2) Find the Nash equilibria on these subgames (3) Redraw the tree replacing the last subgames with the equilibrium payoffs for those subgames. (4) Then move up to the next level of subgames and repeat the procedure. Backward induction Solving for equilibrium by working backward from the end of the game to the beginning - In the subgame when husband following his wife’s choosing Ballet: he would choose Ballet, giving payoffs 2 for her and 1 for him. - In the subgame following his wife’s choosing Boxing, he would choose Boxing, giving payoffs 1 for her and 2 for him. ▪ Next, substitute the husband’s equilibrium strategies for the subgames themselves. ▪ The resulting game is a simple decision problem for the wife, drawn in the lower panel of the figure, a choice between Ballet, which would give her a payoff of 2 and Boxing, which would give her a payoff of 1. ▪ The Nash equilibrium of this game is for her to choose the action with the higher payoff, Ballet. ▪ backward induction allows us to jump straight to the subgame-perfect equilibrium, in which the wife chooses Ballet and the husband chooses “Ballet | Ballet, Boxing | Boxing,” and bypass the other Nash equilibria. Backward induction is particularly useful in games in which there are many rounds of sequential play.  SBA JOE & Sherif Tawfik - 14 Repeated Games Stage (Repeated) game Simple game that is played repeatedly. - Repetition opens up the possibility of the cooperative outcome being played in equilibrium For example, the players in the Prisoners’ Dilemma may anticipate committing future crimes together and thus playing future Prisoners’ Dilemmas together. Players can adopt trigger strategies, whereby they play the cooperative outcome as long as all have cooperated up to that point, but revert to playing the Nash equilibrium if anyone breaks with cooperation. ‫ من بعدها االتني‬,,, ‫اعل‬ ‫ر‬ ‫ ي‬Payoff ‫ علشان ياخد‬Rat ‫يبق‬ ‫ بس اول ما حد فيهم يغش و ي‬, ‫ علطول‬Silent ‫) هيلعبوا‬2 Players) ‫يعن مثل ال‬ ‫ي‬ ‫ علطول‬Rat ‫هيفضلوا يلعبوا‬ - A trigger strategy: One player stops cooperating in order to punish another player for cheating. - One-shot game: A game that is played once. - Tit for tat strategy: Involve only one round of punishment for cheating. - Grim Strategy: Both players will Rat from then on  SBA JOE & Sherif Tawfik - 15 Repeated Games Definite Time Horizon Indefinite Time Horizon ‫زمن محدد‬ ‫أفق ي‬ ‫زمن غت محدد‬ ‫أفق ي‬ Definite Time Horizon For many stage games, repeating them a known, finite number )‫(عدد محدود‬ of times does not increase the possibility for cooperation. Suppose the Prisoners’ Dilemma were repeated for 10 periods. ‫ندهنبيقللناحتنماليةنانهمنيتعاونوانمعن‬,‫نمراتن‬10‫نمثالن‬, ‫ارنبيبقنمحدوحنبعددنمراتنمعين‬ ‫ي‬ ‫هنانالتكر‬ ‫ن‬.(Silent, Silent) (2,2) ‫اللنهون‬ ‫ن ي‬Payoff ‫بعضنويجيبواناعلن‬ ‫ي‬ Indefinite Time Horizon If the number of times the stage game is repeated is indefinite (‫ )غت محدد‬matters change significantly. For example, the partners in crime in the Prisoners’ Dilemma may know that they will participate in many future crimes together, sometimes be caught‫)يتقبضنعليهم(ن‬, and thus have to play the Prisoners’ Dilemma game against each other, but may not know exactly how many opportunities for crime they will have or how often they will be caught. Under certain conditions, more cooperation can be sustained than in the stage game. - Suppose the two players play the following repeated version of the Prisoners’ Dilemma. Let (g) be the probability the game is repeated for another period and (1 – g) the probability the repetitions stop for good. Thus, the probability the game lasts: (at least one period is 1) , (at least two periods is 𝒈), (at least three periods is 𝒈𝟐 ), and so forth. Suppose players use the trigger strategies of playing the cooperative action, Silent, as long a no one cheats by playing Rat, but that players both play Rat forever afterward if either of them had ever cheated. ‫( من بعدها االتني‬3) ‫الل هو‬ ‫ر‬ ‫ ي‬Payoff ‫ و يستفاد بال‬Rat ‫يبق‬ ‫ بس اول ما حد فيهم يغش و ي‬, ‫ علطول‬Silent ‫ هيلعبوا‬2 Players ‫يعن ال‬ ‫ي‬ ‫ علطول‬Rat ‫هيلعبوا‬  SBA JOE & Sherif Tawfik - 16 In equilibrium, both players play Silent and each earns 2 each period the game is played, implying a player’s expected payoff over the course of the entire game is: 2 (1+ 𝒈 + 𝒈𝟐 + 𝒈𝟑 + …) 5.1 If a player cheats and plays Rat, given the other is playing Silent, the cheater earns 3 in that period, but then both play Rat every period, from then on, each earning 1 each period, for a total expected payoff of 3 + 1 ( 𝒈 + 𝒈𝟐 + 𝒈𝟑 + …) 5.2 For cooperation to be a subgame-perfect equilibrium, Equation 5.1 must exceed Equation 5.2. Continuous Actions The new techniques for solving for Nash equilibria will have the same logic as those seen so far. We will illustrate the new techniques in a game called the Tragedy of the Commons. ‫مأساةنالعمو نم‬ Tragedy of the Commons The game involves two shepherds A and B, who graze their sheep on a common (land that can be freely used by community members). Let sA and sB be the number of sheep (‫ )عددناألغنام‬each grazes, Because the common only has a limited amount of space, if more sheep graze, there is less grass for each one, and they grow less quickly. Suppose the benefit (A) gets from each sheep equals: 120 – sA – sB MC = 0 Question one: - Calculate the best response function for each Shepherds - Calculate the Profit for each Shepherds. Answer: - A’s best response will be the number of sheep such that the marginal benefit of an additional sheep equals the marginal cost. His marginal benefit (revenue) of an additional sheep is: MR = 120 – 2sA – sB → Same intercept, Twice the slope Where, MR = MC 120 - 2sA – sB = 0 2SA = 120 – SB SA = 60 – 0.5 SB → A’s best-response function By symmetry, B’s best-response function is: SB = 60 – 0.5 SA ‫نننننن‬  SBA JOE & Sherif Tawfik - 17 Now we can get (SA) SA = 60 – 0.5 SB SA = 60 – 0.5 (60 – 0.5 SA) SA = 60 – 30 + 0.25 SA 0.75 SA = 30 SA = 40 Now we can get (SB) SB = 60 – 0.5 SA → 60 – 0.5 (40) → SB = 40‫ننن‬ For actions to form a Nash equilibrium, they must be best responses to each other. The Nash equilibrium, which lies at the intersection of the two functions, involves each grazing 40 sheep. Best response function ‫ بعملها من ال‬, ‫علشان اعمل الرسمة‬ SA = 60 – 0.5 SB → 60 is intercept Equation ‫ من نفس ال‬SB ‫هجيب ال‬ 0.5 SB = 60 – SA SB = 120 – 2SA → 120 is Intercept Profit (Payoff) for each one = (120 – SA – SB) Q = (120 – 40 – 40) 40 = 1600 What if they each grazed 30 sheep? (120 - SA - SB) Q (120 - 30 - 30) 30 = 1,800 What's the tragedy? Together, they would both be better off by grazing only 30 sheep  SBA JOE & Sherif Tawfik - 18 Incomplete Information Incomplete information: Some players have information about the game that others do not. In all the games studied so far, there was no private information. All players knew everything there was to know about each other’s payoffs, available actions, and so forth. Matters become more complicated, and potentially more interesting, if players know something about themselves that others do not know. Games in which players do not share all relevant information in common are called games of incomplete information. - Screening games: - which include the design of deductible policies by insurance companies in order to deter high-risk consumers from purchasing. - Diplomacy, International Relations ‫الدبلوماسية والعلقات الدولية‬ - Auctions ‫ &نالمزادات‬Card games - Principal-Agent Problem - Signaling games: SBA JOE & Sherif Tawfik - 19 Some MCQ problems SBA JOE & Sherif Tawfik - 20  SBA JOE & Sherif Tawfik - 21 Book problem – Chapter 5 Problem 1: Consider a simultaneous game in which player A chooses one of two actions (Up or Down), and B chooses one of two actions (Left or Right). The game has the following payoff matrix, where the first payoff in each entry is for A and the second for B. A. Find the Nash equilibrium or equilibria. B. Which player, if any, has a dominant strategy? C. Suppose the game is now sequential move, with A moving first and then B. - Write down the extensive form for this sequential-move game. Answer A- If B plays Left → A’s best response is Up If B plays Right → A’s best response is Up IF A Plays UP → B’s best response is Left If A plays Down → B’s best response is Right Nash Equilibrium is when (B Plays Left & A plays Up) → (3,3) B- If B plays Left → A’s best response is Up If B plays Right → A’s best response is Up (A) has a dominant strategy (Plays Up). (B) has no dominant strategy. C- A Up Down B B Left Right Left Right 3,3 5,1 2,2 4,4