Question about Game Theory with Applications in Corporate

Need help with my Game Theory question – I’m studying for my class.

1.Consider the following sequential game played by two players, called Player 1 and Player 2.

(a) What is the normal form of the game?
(b) Find the sub-game perfect Nash equilibrium.
(c) Suppose that Player 2 cannot observe the move of Player 1. Find the perfect Bayesian equilibrium.

2. Consider. the following game of incomplete information played by two players. In this game, each player has 2 strategies, called strategy 1 and strategy 2. Furthermore, Player 1 knows the preferences of Player 2, but Player 2 does not know the preferences of Player 1. For simplicity, suppose that Player 1 has two types, say a and b. Furthermore, it is common knowledge that there is a 50-50 chance that Player 1 can be of either type. The payoff matrices are as follows:

Games with Incomplete Information 1. A Simple Game with Incomplete Information John is in love with Mary, but he does not know whether Mary likes him or not. From his own experience with Mary, he thinks there is a chance of 25% that mary likes him, and a chance of 75% that Mary does not like him. John is thinking of inviting Mary to dinner. The utility of John depends on (i) his strategy – invite or do not invite – (ii) the responese of Mary when asked – accept or refuse the invitation – and (iii) the type of Mary (whether she likes him or not). As for Mary, her utility also depends on (i) the strategy chosen by John – invite or do not invite – (ii) her response when asked – accept or not accept – and (iii) her type (whether she likes him or not). The following two matrices represent the payoffs for John and Mary, as functions of the type of Mary and the combination of strategies chosen by the two individuals. Mary 1 (0.25) Mary 2 (0.75) (Mary likes John) (Mary does not like John) Accept Refuse Accept Refuse Invite (10, 7) (2, 2) (8, 1) (2, 4) John Do not invite (3, 4) (3, 4) (3, 4) (3, 4) In the above matrices, Mary 1 represents the type <> and Mary 2 represents the type <>. We would like to find the answers to the following questions: (i) Should John invite Mary to dinner? (ii) Should Mary accept the offer if she likes him? (iii) Should Mary accept the offer if she does not like him? The game just described is called as game with incomplete information because John does not know the type of Mary, i.e., he does not know whether he is facing a girl who likes him or a girl who does not like him. In the two payoff matrices, Mary 1 represents the type Mary likes John, and Mary 2 represents the type Mary does not like John. 1.1. The Solution of the Game: the Bayesian Nash Equilibrium Obviously, Mary knows whether she likes John or not, i.e., she knows her own type. If she likes him, then she knows that the first matrix represents the payoff matrix for the game between John and her. On the other hand, if Mary does not like John, then from her perspective, it is the second matrix that represents the game between John and her. John, on the other hand, because he does not know the type of Mary, thinks that the first matrix applies with probability 0.25, and the second matrix applies with probability 0.75. In his decision making, he has to take into account both of these possibilities. Strictly speaking, the game being analyzed can be considered as a game that involves three players: John, Mary 1, and Mary 2. Mary 1 knwos that she is playing the game represented by the first payoff matrix, while Mary 2 knows that she plays the game represented by the second payoff matrix. As for John, he does not know exactly which game he is playing. He thinks that he play the first game with probability 0.25 and the second game with probability 0.75. For Mary 1, the strategy <> gives her a payoff of 7 if John invites her to dinner, and a payoff of 4 if John does not invite her to dinner. On the other hand, if Mary 1 plays the strategy <>, then she obtains a payoff of 2 if John invites her to dinner and a payoff of 4 if John does not invite her to dinner. Thus <> is the dominant strategy for Mary 1, and she will play <>. For Mary 2, the strategy <> gives her a payoff of 1 if John invites her to dinner, and a payoff of 4 if John does not invite her to dinner. On the other hand, if Mary 2 plays the strategy <>, then she obtains a payoff of 4 if John invites to dinner and a payoff of 4 if John does not invite her to dinner. Thus <> is the weakly dominant strategy for Mary 1, and she will play <>. For John, according to the first-order rationality principle, he knows that Mar 1 will play <>, and Mary 2 will play <>. If John plays <>, then there is 25% chance that he is playing against Mary 1 in which case she will acceplt the offer, giving him a payoff of 10. On the other hand, because there is a 75% chance that it is Mary 2 that he is asking to dinner, and he knows that Mary 2 will refuse, with the ensuing consequence that he will obtain a payoff of 2. Thus playing <,Invite>> gives John an expected payoff of (1) 0.25×10 + 0.75×2 = 4.0 On the other hand, if John plays <>, Mary 1 will play <>, which is the dominant strategy for her, and the payoff for John is then 3. For Mary 2, she plays <>, and John’s payoff is 3. Thus, playing <> gives John a payoff of 3. Because <> yields John a higher payoff (4) than <> (3), John will invite Mary to dinner. As for Mary, she accepts the offer if she likes John, but turns down the offer if she does not like John. This equilibrium is called a Bayesian Nash equilibrium of the game with incomplete information. Under the Bayesian Nash equilibrium, the expected payoff for John is 4. For Mary 1, her payoff is 7. For 2 ��� Games with Incomplete Information.nb 1.2. The Value of Perfect Information Suppose that the little sister of Mary knows whether Mary likes John or not, and is willing to provide John with this information at a price. What is that price? If John accepts such a proposition, and if Mary likes him, then this information will be given to him by the little sister. In this case, and the probability of this event is 0.25, he will invite mary out to dinner because <> is the dominant strategy of Mary 1. This event gives him a payoff of 10. On the other hand, if Mary does noit like him, then with this information – which occurs with probability 0.75 – he will not invite Mary out to dinner because he knows that <> is weakly dominant for Mary 2. The payoff John obtains under this event is then equal to 3. Thus, the expected payoff for John if he accepts the proposition of Mary’s little sister is (2) 10×0.25 + 3×0.75 = 4.75. The difference between (2) and (1), namely (3) 4.75 – 4 = 0.75 is the gain in expected payoff that John obtains if he accepts the proposition of Mary’s sister, and this gain is the maximum amount John is willing to pay for the information. The payoff differential 0.75 represents the value of perfect information on the type of Mary. 2. A More Complicated Games with Incomplete Information: A Two-Person Game in which Each Player has Two Types In this game, there are two players: player I and player II. Player I has two types a and b, while player II has two types c and d. A player knows his type, but not the type of the other player. Thus, in reality there are 4 possible games: (i) The game between Ia and IIc, (ii) The game between Ia and IId, (iii) The game between Ib and IIc, (iv) The game between Ib and IId. In game (i) player I knows that his type is a, but does not know whther he plays against IIc or IId. On the other hand, in this game, player II knows that his type is c, but does not know whether he is paying against Ia or Ib. The interpretation of games (ii), (iii), and (iv) are similar. The game evolves through time as follows. Nature moves first, and makes a random choice among the following set of possible type profiles: {(Ia, IIc), (Ia, IId), (Ib, IIc), (Ib, IId)}. The following matrix represents the probabilities of the four possible type profiles: Games with Incomplete Information.nb ���3 Player I Ia Ib IIc IId Player II 1 100 0 90 100 9 100 According to the above matrix, Nature chooses the type profile (Ia, IIc) with probability 1 100 and the type profile (Ia, IId) with probability 0. In the same manner, Nature chooses the type profile (Ib, IIc) with probability 9 100 and the type profile (Ib, IId) with probability 90 100 . If the random choice made by Nature is (Ia, IIc), Nature reveals secretly to player I that his type is Ia et secretly to player II that his type is IIc. However, Ia does not know whether he is playing against IIc or IId. In the same manner, IIc does not know whether he is playing against Ia or Ib. Each player has 2 strategies, say 1 and 2. The following 4 matrices represent the payoffs for the two players as functions of their types and their strategies: IIc IId 1 2 1 2 Ia 1 (8,-8) (-8,8) (-4,4) (8,-8) 2 (0,0) (-4,4) (0,0) (12,-12) Ib 1 (-8,8) (12,-12) (4,-4) (-4,4) 2 (-12,12) (16,-16) (0,0) (-8,8) 2.1. The Solution of the Game: The Bayesian Nash Equilibrium 4 ��� Games with Incomplete Information.nb Applying the Bayes’ formula, we obtain p[c, a] = Prob {IIc Ia} = 1 100 1 100+0 = 1, p[d, a] = Prob {IId Ia} = 0 1 100+0 = 0, p[c, b] = Prob {IIc Ib} = 9 100 9 100+ 90 100 = 1 11 , p[d, b] = Prob {IId Ib} = 90 100 9 100+ 90 100 = 10 11 , p[a, c] = Prob {Ia IIc} = 1 100 1 100+ 9 100 = 1 10 , p[b, c] = Prob {Ib IIc} = 9 100 1 100+ 9 100 = 9 10 , p[a, d] = Prob {Ia IId} = 0 90 100 = 0, p[b, d] = Prob {Ib IId} = 90 100 0+ 90 100 = 1. We shall carry out the computations with the help of Mathematics. First, we enter the data on the payoff matrices. Because in the four payoff matrices, the gain of one player is the opposite of the gain of the other player, we only need to enter the payoff for one player, say player I. The following matrix gives the gain for Ia when he plays against Ic. ������� m[a, c] = {{8, -8}, {0, -4}} ������� {{8, -8}, {0, -4}} ������� MatrixForm[m [a, c]] �������������������  8 -8 0 -4  ������� m[a, c] // MatrixForm �������������������  8 -8 0 -4  The payoff matrix when Ia plays against IId ������� m[a, d] = {{-4, 8}, {0, 12}} ������� {{-4, 8}, {0, 12}} ������� m[a, d] // MatrixForm �������������������  -4 8 0 12  The payoff matrix when Ib plays against IIc Games with Incomplete Information.nb ���5 ������� m[b, c] = {{-8, 12}, {-12, 16}} ������� {{-8, 12}, {-12, 16}} ������� m[b, c] // MatrixForm �������������������  -8 12 -12 16  The payoff matrix when Ib plays against IId ������� m[b, d] = {{4, -4}, {0, -8}} ������� {{4, -4}, {0, -8}} ������� MatrixForm[m[b, d]] �������������������  4 -4 0 -8  The conditional probailities that represent the beliefs of one player concerning the type of the other player. �������� p[c, a] = 1 �������� 1 �������� p[d, a] = 0 �������� 0 �������� p[c, b] = 1 11 �������� 1 11 �������� p[d, b] = 10 11 �������� 10 11 �������� p[a, c] = 1 10 �������� 1 10 �������� p[b, c] = 9 10 �������� 9 10 �������� p[a, d] = 0 �������� 0 6 ��� Games with Incomplete Information.nb �������� p[b, d] = 1 �������� 1 The game between the two players I and II in which each of them has 2 possible types can be considered as a game with 4 players: Ia, Ib, IIc, IId, where each type of each player can be considerd as a distinct player. However, the computations of the expected payoff for each type of each player require some particular attention. For example, in the game between (Ia, IIc), player Ia knows that Ib is not involved and that player Ia does not know against whom he is playing: IIc or IId. For player Ia, player Ib does not appear in the game he is playing. Similarly, player IIc knows that player IId is not in the game, but player Ia or player Ib might be his adversary with probability p[a, c] and p[b, c], respectively. A combination of strategies for the four players Ia, Ib, IIc, IId can be represented by a list of positive integers: {i, j, k, ℓ}, i = 1, 2, j = 1, 2, k = 1, 2, ℓ = 1, 2, where i is the strategy chosen by player Ia; j is the strategy chosen by player Ib; k is the strategy chosen by player IIc; and ℓ is the strategy chosen by player IId. For example, the list {1, 1, 1, 1} represents the combination of the following strategies: (i) player Ia chooses strategy 1; (ii) player Ib chooses strategy 1; (iii) player IIc chooses strategy 1; (iv) player IId chooses strategy 1. As another example, the list {2, 1, 1, 2} represents the combination of the following strategies: (i) player Ia chooses strategy 2; (ii) player Ib chooses strategy 1; (iii) player IIc chooses strategy 1; (iv) player IId chooses strategy 2. Under the list {1, 1, 1, 1}, player Ia chooses strategy 1. Player Ia plays against IIc with probability p[c, a] and against player IId with probability p[d, a]. Because under the list {1, 1, 1, 1}, player IIc chooses strategy 1, the payoff obtained by player Ia is 8, and this event has probability p[c, a]. Also, because under the list {1, 1, 1, 1}, player IId chooses strategy 1, the payoff obtained by player Ia is -4, and this event has probability p[d, a]. Thus, the expected payoff for player Ia under the combination of strategy {1, 1, 1, 1} is (1) 8 p[c, a] + (-4) p[d, a] = 8×1 + (-4)×0 = 8. Under the list {1, 1, 1, 1}, player Ib chooses strategy 1. Player Ib plays against IIc with probability p[c, b] and against player IId with probability p[d, b]. Because under the list {1, 1, 1, 1}, player IIc chooses strategy 1, the payoff obtained by player Ib is -8, and this event has probability p[c, b]. Also, because under the list {1, 1, 1, 1}, player IId chooses strategy 1, the payoff obtained by player Ib is 4, and this event has probability p[d, b]. Thus, the expected payoff for player Ib under the combination of strategy {1, 1, 1, 1} is Games with Incomplete Information.nb ���7 (2) -8 p[c, b] + 4 p[d, b] = -8× 1 11 + 4× 10 11 = 32 11 . Under the list {1, 1, 1, 1}, player IIc chooses strategy 1. Player IIc plays against Ia with probability p[a, c] and against player Ib with probability p[b, c]. Because under the list {1, 1, 1, 1}, player Ia chooses strategy 1, the payoff obtained by player IIc is -8, and this event has probability p[a, c]. Also, because under the list {1, 1, 1, 1}, player Ib chooses strategy 1, the payoff obtained by player IIc is 8, and this event has probability p[b, c]. Thus, the expected payoff for player IIc under the combination of strategy {1, 1, 1, 1} is (3) -8 p[a, c] + 8 p[b, c] = -8× 1 10 + 8× 9 10 = 64 10 = 32 5 . Under the list {1, 1, 1, 1}, player IId chooses strategy 1. Player IId plays against Ia with probability p[a, d] and against player Ib with probability p[b, d]. Because under the list {1, 1, 1, 1}, player Ia chooses strategy 1, the payoff obtained by player IId is 4, and this event has probability p[a, d]. Also, because under the list {1, 1, 1, 1}, player Ib chooses strategy 1, the payoff obtained by player IId is -4, and this event has probability p[b, d]. Thus, the expected payoff for player IId under the combination of strategy {1, 1, 1, 1} is (4) 4 p[a, d] + -4 p[b, d] = 4×0 + (-4)×1 = -4. Putting (1)-(4) together, we obtain the following list of expected payoffs for the four players {Ia, Ib, IIc, IId} : (5) {1, 1, 1, 1} → 8, 32 11 , 32 5 , -4. The computations needed to obtain (5) are laborious. Because there are 4 players and each player has two possible strategies, there are in total 16 combinations of strategies (6) {i, j, k, ℓ}, i = 1, 2, j = 1, 2, k = 1, 2, ℓ = 1, 2. Now we write a Mathematica program to obtain the lists of payoffs for all the possible combinations of strategies (6). �������� u[i_, j_, k_, ℓ_] := {p[c, a] × m[a, c][[i, k]] + p[d, a] × m[a, d][[i, ℓ]], p[c, b] × m[b, c][[j, k]] + p[d, b] × m[b, d][[j, ℓ]], -p[a, c] × m[a, c][[i, k]] – p[b, c] × m[b, c][[i, k]], -p[a, d] × m[a, d][[i, ℓ]] – p[b, d] × m[b, d][[j, ℓ]]} Using the function just defined, we obtain the following list of expected payoffs for {Ia, Ib, IIc, IId} under the combination of strategies {1, 1, 1, 1} : �������� u[1, 1, 1, 1] �������� 8, 32 11 , 32 5 , -4 The list of expected payoffs for {Ia, Ib, IIc, IId} under the combination of strategies {1, 2, 1, 1} : �������� u[1, 2, 1, 1] �������� 8, – 12 11 , 32 5 , 0 8 ��� Games with Incomplete Information.nb The list of vectors of payoffs for (Ia, Ib, IIc, IId) that correspond to all possible combinations of strategies {i, j, k, ℓ} : �������� t[0] = Table[{{i, j, k, ℓ}, u[i, j, k, ℓ]}, {i, 1, 2}, {j, 1, 2}, {k, 1, 2}, {ℓ, 1, 2}] �������� {1, 1, 1, 1}, 8, 32 11 , 32 5 , -4, {1, 1, 1, 2}, 8, – 48 11 , 32 5 , 4, {1, 1, 2, 1}, -8, 52 11 , -10, -4, {1, 1, 2, 2}, -8, – 28 11 , -10, 4, {1, 2, 1, 1}, 8, – 12 11 , 32 5 , 0, {1, 2, 1, 2}, 8, – 92 11 , 32 5 , 8, {1, 2, 2, 1}, -8, 16 11 , -10, 0, {1, 2, 2, 2}, -8, – 64 11 , -10, 8, {2, 1, 1, 1}, 0, 32 11 , 54 5 , -4, {2, 1, 1, 2}, 0, – 48 11 , 54 5 , 4, {2, 1, 2, 1}, -4, 52 11 , -14, -4, {2, 1, 2, 2}, -4, – 28 11 , -14, 4, {2, 2, 1, 1}, 0, – 12 11 , 54 5 , 0, {2, 2, 1, 2}, 0, – 92 11 , 54 5 , 8, {2, 2, 2, 1}, -4, 16 11 , -14, 0, {2, 2, 2, 2}, -4, – 64 11 , -14, 8 The list of vectors of payoffs for (Ia, Ib, IIc, IId) that correspond to all possible combinations of strategies {i, j, k, ℓ} has dimension 2×2×2×2. An element of this list, say {1, 1, 1, 1}, 8, 32 11 , 32 5 , -4 contains two lists. The first list {1, 1, 1, 1} represents the combination of strategies chosen by the four players Ia, Ib, IIc, IId, and the second list 8, 32 11 , 32 5 , -4 gives the expected payoff for the four players under the combination of strategies {1, 1, 1, 1}. To make t[0] more readable, we now suppress the unnecessary pairs of parentheses. �������� t[1] = Flatten[t[0], 3] �������� {1, 1, 1, 1}, 8, 32 11 , 32 5 , -4, {1, 1, 1, 2}, 8, – 48 11 , 32 5 , 4, {1, 1, 2, 1}, -8, 52 11 , -10, -4, {1, 1, 2, 2}, -8, – 28 11 , -10, 4, {1, 2, 1, 1}, 8, – 12 11 , 32 5 , 0, {1, 2, 1, 2}, 8, – 92 11 , 32 5 , 8, {1, 2, 2, 1}, -8, 16 11 , -10, 0, {1, 2, 2, 2}, -8, – 64 11 , -10, 8, {2, 1, 1, 1}, 0, 32 11 , 54 5 , -4, {2, 1, 1, 2}, 0, – 48 11 , 54 5 , 4, {2, 1, 2, 1}, -4, 52 11 , -14, -4, {2, 1, 2, 2}, -4, – 28 11 , -14, 4, {2, 2, 1, 1}, 0, – 12 11 , 54 5 , 0, {2, 2, 1, 2}, 0, – 92 11 , 54 5 , 8, {2, 2, 2, 1}, -4, 16 11 , -14, 0, {2, 2, 2, 2}, -4, – 64 11 , -14, 8 To find the Bayesian Nash equilibrium, we go through the list t[1] one element at a time. Games with Incomplete Information.nb ���9 First, consider the list {1, 1, 1, 1}, 8, 32 11 , 32 5 , -4. If player Ia switches from strategy 1 to strategy 2, then the new combination of strategies that now applies is {2, 1, 1, 1}, and the list of expected payoffs for the 4 players is {2, 1, 1, 1}, 0, 32 11 , 54 5 , -4. Furthermore, the expected payoff for player Ia falls from 8 to 0, and this means that player Ia will not change strategy when players Ib, IIc, IId all choose strategy 1. Thus, strategy 1 is the best response of player Ia when players Ib, IIc, IId all choose strategy 2. Similarly, starting from the combination of strategies {1, 1, 1, 1}, if player Ib switches to strategy 2, then the list of ecpected payoffs becomes {1, 2, 1, 1}, 8, -12 11 , 32 5 , 0. The expected payoff for player Ib falls from 32 11 to -12 11 , and this means that player Ib will not deviate, either. Also, starting from the combination of strategies {1, 1, 1, 1}, if player IIc switches to strategy 2, then the list of ecpected payoffs becomes {1, 1, 2, 1}, -8, 52 11 , -10, -4 The expected payoff for player IIc falls from 32 5 to -10, and this means that player IIc will not deviate, either. Finally, starting from the combination of strategies {1, 1, 1, 1}, if player IId switches to strategy 2, then the list of ecpected payoffs becomes {1, 1, 1, 2}, 8, -48 11 , 32 5 , 4 The expected payoff for player IId rises from -4 to 4, and this means that player IId will deviate. We have just shown that the combination of strategies (1, 1, 1, 1} is not a Nash equilibrium. Next, consider the combination of strategies {1, 1, 1, 2}, 8, -48 11 , 32 5 , 4. If player Ia chnages strategy, then the new combination of strategies that applies is {2, 1, 1, 2}, 0, – 48 11 , 54 5 , 4 and this action leads to a fall in his expected payoff from 8 to 0. Thus, player Ia will not deviate. If player Ib chnages strategy, then the new combination of strategies that applies is {1, 2, 1, 2}, 8, -92 11 , 32 5 , 8, and this action leads to a fall in his expected payoff from -48 11 to -92 11 . Thus, player Ib will not deviate. If player IIc chnages strategy, then the new combination of strategies that applies is {1, 1, 2, 2}, -8, -28 11 , -10, 4 10 ��� Games with Incomplete Information.nb and this action leads to a fall in his expected payoff from 32 5 to -10. Thus, player IIc will not deviate. If player IId chnages strategy, then the new combination of strategies that applies is {1, 1, 1, 1}, 8, 32 11 , 32 5 , -4 and this action leads to a fall in his expected payoff from 4 to -4. Thus, player IId will not deviate. The combination of strategies {1, 1, 1, 2} thus constitutes a Bayesian Nash equilibrium. One can verify (with a lot of hard work) that the remaining combinations of strategies, just like {1, 1, 1, 1}, are not Nash equilibria. 2.2. The Bayesian Nash equilibrium of the game The game has a unique Bayesian Nash equilibrium in pure strategies, which is given by {1, 1, 1, 2}, 8, -48 11 , 32 5 , 4. There might exist Bayesian Nash equilibria in mixed strategies. It remains to be found