经济博弈论期末考试答案_2010

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(b) How many strategies does each player have? Show the game in strategic form, where the payoffs in each cell reflect the expected payoffs given each player’s respective strategy. (3 pts) Felix has two actions at each of two nodes, so he has 2*2 = 4 strategies. Oscar has two actions at the single information set, so he has two strategies. The strategic form is:
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经济博弈论（2010 年春季学期） 期末考试答案 (2010/6/9) 1、 （20 分）菲利克斯和奥斯卡正在玩一个简化版的扑克游戏，名为“脱光扑克” 。双方一开 始都下 1 美元的注。菲利克斯（且只有菲利克斯）取牌，取的牌要么是 K 要么是 Q，二者 等概率（因为一共有四张 K 和四张 Q） 。菲利克斯随后选择是退出还是赌博。如果菲利克斯 选择退出，游戏结束，奥斯卡得到菲利克斯的 1 美元。如果菲利克斯选择赌博，他加注 1 美元，随后奥斯卡选择是退出还是叫牌。 如果奥斯卡选择退出，菲利克斯赢得奥斯卡的 1 美元。如果奥斯卡叫牌，他也需要加 注 1 美元，使其赌注与菲利克斯的相等。然后菲利克斯亮牌。如果牌是 K，则菲利克斯赢得 奥斯卡的 2 美元赌注。如果牌是 Q，奥斯卡赢得菲利克斯的 2 美元赌注。 (a) 用扩展式表示该博弈（注意信息集） 。 （5 分） (b) 每个参与者有多少个策略？用策略式表示该博弈，其中每一格中的收益反映每个参与者 相应策略下的预期收益 。 （3 分） ．．．． (c) 消去劣势策略 （如有） 。 找出博弈的混合策略均衡。 均衡中菲利克斯的预期收益是多少？ （3 分） (d) 在博弈的扩展式中，如果双方都出上述的纳什均衡策略，求出在信息集中相应参与者的 信念（即该参与者觉察到的他处在信息集中每一个节点上的概率） 。 （4 分） (e) 描述该博弈的贝叶斯完美均衡。该博弈会有几个贝叶斯完美均衡？解释你的回答。 （5 分） (20 points) Felix and Oscar are playing a simplified version of poker called Stripped-Down Poker. Both make an initial bet of one dollar. Felix (and only Felix) draws on card, which is either a King or a Queen with equal probability (there are four Kings and four Queens). Felix then chooses whether to Fold or to Bet. If Felix chooses to Fold, the game ends, and Oscar receives Felix’s dollar in addition to his own. If Felix chooses to Bet, he puts in an addition dollar, and Oscar chooses whether to Fold or to Call. If Oscar Folds, Felix wins the pot (consisting of Oscar’s initial bet of one dollar and two dollars from Felix). If Oscar Calls, he puts in another dollar to match Felix’s bet, and Felix’s card is revealed. If the card is a King, Felix wins the pot (two dollars from each of the roommates). If it is a Queen, Oscar wins the pot. (a) Show the game in extensive form. (Be careful about information sets.) (5 pts) The extensive form of Stripped-down Poker is:
The payoffs in the table are the expected payoffs to the players. For example, if Felix plays BF and Oscar plays Call, with probability 0.5 Felix receives a king and Bets. After this play Oscar calls, for payoffs of (2, –2). And with probability 0.5 Oscar receives a Queen and folds, for payoffs of (–1, 1). The expected payoffs in the cell (BF, Call) are thus 0.5 * (2, –2) + 0.5 (–1, 1) = (0.5, –0.5). We find the payoffs of the rest of the table the same way. (c) Eliminate dominated strategies, if any. Find the equilibrium in mixed strategies. What is the expected payoff to Felix in equilibrium? (3 pts) FB is dominated by BB, and FF is dominated by BF. The reduced game table, with best responses underlined, is: Oscar Felix Bet if K, Bet if Q (BB) Call 0, 0 Fold 1, – 1
2、 （20 分）考虑一个在父母和孩子之间进行的（两人）博弈。孩子可以选择乖（G）和不乖从不乖中得到价值为 1 的快乐，但 如遇惩罚则受到价值为-2 的损害。也就是说，如果孩子表现乖而不受惩罚则得到 0；如果表 现不乖但受到惩罚则得 1-2=-1；以此类推。父母从孩子不乖的行为中得到-2，从实施惩罚中 得到-1。 (a) 将这一博弈建立为一个同时博弈，并找出其均衡。 （4 分） (b) 现在假定孩子先选择 G 或 B，父母在观察到孩子的行动后再选择 P 或 N。画出博弈树， 找出其子博弈完美均衡。 （4 分） (c) 现在假定在孩子行动之前，父母可以对其策略做出承诺，例如威胁“P 如果 B” （ “如果 你表现不乖，我将惩罚你” ） 。父母有多少种这样的策略？将这些策略用策略性行动—— 威胁、许诺或承诺——命名（如果可以的话） 。 （4 分） (d) 画出这一博弈的博弈表，找出所有纯策略纳什均衡。 （4 分） (e) 是否存在某种策略性行动可以使得父母比在同时博弈（(a)问）中更好？解释你的答案。 （4 分） (20 points) Consider a game between a parent and a child. The child can choose to be good (G) or bad (B); the parent can punish the child (P) or not (N). The child gets enjoyment worth a 1 from bad behavior, but hurt worth -2 from punishment. Thus a child who behaves well and is not punished gets a 0; one who behaves badly and is punished gets 1-2=-1; and so on. The parent gets -2 from the child’s bad behavior and -1 from inflicting punishment. (a) Set up this game as a simultaneous-move game, and find the equilibrium. (4 pts)
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Bet if K, Fold if Q (BF)
0.5, – 0.5
0, 0
The game has no equilibrium in pure strategies, but it has a mixed-strategy Nash equilibrium where Felix plays BB with probability 1/3 and Oscar Calls with probability 2/3. The expected payoff to Felix in equilibrium is a weighted average of the four possible payoffs in the smaller game table, where the weights are given by the products of the probabilities with which the two players’ strategies are played in equilibrium: (1/3)*(2/3)*(0) + (1/3)*(1/3)*(1) + (2/3)*(2/3)*(1/2) + (2/3)*(1/3)*(0) = 1/3 (d) If both players play the Nash equilibrium strategies solved above, in the extensive form, show the beliefs of corresponding player on the information set (i.e., probabilities on each node in the information set that player perceives he or she will have). (4 pt) Oscar’s beliefs on the information set are: prob=(1/2)/(1/2+1/2*(1/3))=3/4 for B from K, and prob=1/4 for B from Q. (e) Describe the perfect Bayesian equilibrium (or equilibria) of the game. How many Perfect Bayesian equilibria do you think this game will have? Explain. (5 pts) Since PBE is just a refinement of Nash equilibrium, and we can verify the Nash equilibrium solved in part (c) can be support as a PBE, with beliefs in part (d). So the only PBE is as: (i) Felix plays B when Nature chooses K, and plays B and F with probability 1/3 and 2/3 otehrwise. (ii) Oscar plays F and C with probability 1/3 and 2/3. (iii) Oscar’s beliefs on his/her information set are: prob= 3/4 for B from K, and prob=1/4 for B from Q.