DYNAMIC GAMES with incomplete information. Lecture 11

Transcription

1 DYNAMIC GAMES with incomplete information Lecture

2 Revision Dynamic game: Set of players: A B Terminal histories: 2 all possible sequences of actions in the game Player function: function that assigns a player to every proper subhistory Preferences for the players: Preferences over terminal histories, X, C Y D, 2 represented by utility (payoff) function GAME THEORY 29/2

3 Incomplete inform. Dynamic games NOW we allow for some level of uncertainty also in dynamic games We will start with the simple example: (Variant of Bach or Stravinsky) Two people wish to go out together. Two concerts are available: one of music by Bach, and one of music by Stravinsky. One person prefers Bach and the other prefers Stravinsky. If they go to different concerts, each of them is equally unhappy listening to the music of either composer. But now, they are choosing the concert sequentially. After the first person make a choice, the second person receives a signal (information) about the choice of first person and makes her choice according to this signal. GAME THEORY 29/2

4 B or S: perfect information If the signal that the second person receives has different values after P plays Bach and after P plays Stravinsky, the second person is perfectly informed about the P s choice and the game is dynamic game with P perfect information. P2 P2,,, 2 GAME THEORY 29/2

5 B or S: imperfect information If the signal that the second person receives has same value after P plays Bach and after P plays Stravinsky, the second person is not informed about the P s choice and the game is the dynamic game with imperfect information. We denote the information that P2 has by the information set (dashed red line) P2 P P2,,, 2 GAME THEORY 29/2

6 B or S: imperfect information information set (dashed red line) collection of decision nodes (histories after which it is player s turn) such that: a) when the play reaches a node in the information set, the player P with the move does not know which node in the information set has been reached. b) The player has the move and same set of choices at each node in the Information set. P2 P2,,, 2 GAME THEORY 29/2

7 Dynamic games Incomplete inform. Set of players Terminal histories: all possible ways(sequences of actions) how we can get at some ending node in the tree diagram Player function: function that assign to each history (not terminal history) either player or chance A function that assign to each history after which it is chance turn a probability distribution over the actions available after that history Information partition division of histories (decision nodes) of the player that has the turn into information sets Preferences for the players preferences over the set of lotteries over terminal histories represented by the vnm preferences (expected utility theory) GAME THEORY 29/2

8 Dynamic games Incomplete inform. Set of players: person, person 2 Terminal histories: {B,B}, {B,S}, {S,B}, {S,S} Player function: P(ø)=; P(B)= P(S)=2 Chance - None Information partition Player s information partition contains one information set {ø}, Player 2 s information partition contains also one information set {B,S}. (in the case of perfect information, P2 s information partition contains two information sets {B}, {S}) Preferences for the players represented by utility function (payoffs) GAME THEORY 29/2

9 Example P RIGHT 4 P2 P2,, 4,, 2 GAME THEORY 29/2

10 Example Set of players: person, person 2 Terminal histories: {LEFT,LEFT}, {LEFT,RIGHT}, {MIDDLE,LEFT}, {MIDDLE,RIGHT}, {RIGHT} Player function: P(ø)=; P(LEFT)= P(MIDDLE)=2 Chance - None Information partition Player s information partition contains one information set {ø}, Player 2 s information partition contains two information set {LEFT,MIDDLE}, {RIGHT} Preferences for the players represented by utility function (payoffs) GAME THEORY 29/2

11 Example 2: Simple poker game Two people are playing a following card game each of them having just 2 dollars: At the beginning of the game each player has to put one dollar into the pot (mandatory bet). Then the first player (dealer) draws a card from a deck which contains only KINGS and QUEENS. With probability.5 player draws KING and with probability.5 player draws QUEEN. After the player privately observes her own card, she moves by either FOLDING or RAISING. FOLD means that the game ends and player lose one dollar, player 2 earns one dollar. RAISE means that she adds an additional dollar to the pot. After RAISE 2 nd player either FOLD (loosing one dollar) or CALL (add additional dollar to the pot). Folding ends the game. If the 2 nd player CALLs player wins the pot is she has KING and loose if she has QUEEN. GAME THEORY 29/2

12 RAISE RAISE Example 2: Simple poker game Chance FOLD FOLD P -, -, P P2 P2, - - 2, - -2 GAME THEORY 29/2

13 Example 2: Simple poker game Set of players: player, player (chance) Terminal histories: {Queen,Fold}, {King,Fold}, {Queen,Raise,Fold}, {Queen,Raise,Call}, {King,Raise,Fold}, {King,Raise,Call}, Player function: P(ø)=Chance; P(Queen)=, P(King)=; P(Queen, Raise)= P(King, Raise)=2 Chance Queen = ½ ; King = ½ Information partition Player s information partition contains two information set {Queen}, {King} Player 2 s information partition contains two information set {{Queen,Raise},{King,Raise}}, {{Queen,Fold},{King,Fold}},. Preferences for the players represented by utility function (payoffs) GAME THEORY 29/2

14 Example : Signaling games Signaling game is a dynamic game of incomplete information involving two players: a Sender (S), and a Receiver (R). The timing of the game is as follows: ) Chance (Nature) draws a type t i for the Sender from a set of feasible types T={t, t N } according to probability distribution {p,, p N } such that p + +p N = 2) The sender observes her type and then chooses a message (signal) m i from a set of feasible messages M={m, m J } ) The Receiver observes m j (but not t i ) and then chooses an action a k from a set of feasible actions A={a, a K } GAME THEORY 29/2

15 Example : Signaling game 2 types Signaling game is a dynamic game of incomplete information involving two players: a Sender (S), and a Receiver (R). The timing of the game is as follows: ) Chance (Nature) draws a type t i for the Sender from a set of feasible types T={X, Y} according to probability distribution such that p X +p Y = (p X =p; p Y =-p) 2) The sender observes her type and then chooses a message (signal) m i from a set of feasible messages M={High,Low} ) The Receiver observes m j (but not t i ) and then chooses an action a k from a set of feasible actions A={Left,Right} GAME THEORY 29/2

16 -p p Type Y Type X Signaling game 2 types, P2 HIGH P LOW P2 4, Chance, 4, P2 HIGH P LOW P2,, 2 GAME THEORY 29/2

17 Example : Signaling game 2 types Set of players: player, player (chance) Terminal histories: {TypeX,High,Left}, {TypeX,High,Right}, {TypeX,Low,Left}, {TypeX,Low,Right}, {TypeY,High,Left}, {TypeY,High,Right}, {TypeY,Low,Left}, {TypeY,Low,Right}, Player function: P(ø)=Chance; P(TypeX)=, P(TypeY)=; P(TypeX, High)= P(TypeX, Low)= P(TypeY, High)= P(TypeY, Low)= Chance TypeX = p ; TypeY = -p Information partition Player s information partition contains two information set {TypeX}, {TypeY} Player 2 s information partition contains also two information set {{TypeX,High},{TypeY,High}}, {{TypeX,Low},{TypeY,Low}} GAME THEORY 29/2

18 Information set definition: collection of decision nodes (histories after which it is player s turn) such that:. when the play reaches a node in the information set, the player with the move does not know which node in the information set has been reached. 2. The player has the move and same set of choices at each node in the Information set GAME THEORY 29/2

19 Belief system We assume that if an information set that contains more than one history, the player whose turn it is to move forms a belief about the history that has occurred. We model this belief as a probability distribution over the histories in the information set. p A p B p C p X p Y p Z p A +p B +p C = p X +p Y = p Z = GAME THEORY 29/2

20 Belief system Definition: Belief system assigns to each information set a probability distribution over the decision nodes (histories) in that information set. p A p B p C p X p Y p Z p A +p B +p C = p X +p Y = p Z = GAME THEORY 29/2

21 Behavioral strategy In the case of dynamic games with perfect information, the strategy defines action at every node where it is the player s turn. However, now the player does not know the exact node (history), but just the information set. p A p B p C p X p Y p Z BEHAVIORAL STRATEGY: PLAN OF ACTION - assigns action for each information set at which it is the player s turn GAME THEORY 29/2

22 Behavioral strategy To incorporate both pure and mixed strategies: BEHAVIORAL STRATEGY: assigns to each information set at which it is the player s turn a probability distribution over all feasible actions of the player in that information set. p A p B p C p X p Y p Z S L S R T L T R U L U R BEHAVIOR STRATEGY: set of 2 probabilities for each information set, such that S L +S R = T L +T R = U L +U R = GAME THEORY 29/2

23 Weak Sequential Equilibrium More complex and richer games strengthening equilibrium concept. Static games with complete information Nash equilibrium 2. Dynamic games with complete information Subgame perfect Nash equilibrium. Static games with incomplete information Bayesian Nash equilibrium 4. Dynamic games with incomplete information weak sequential equilibrium - refines Bayesian Nash equilibrium in the same sense as Subgame perfect Nash equilibrium refines Nash equilibrium Introduces sequential rationality into Bayesian Nash equilibrium GAME THEORY 29/2

24 Weak Sequential Equilibrium Definition: A weak sequential equilibrium consists of behavioral strategies and beliefs systems satisfying following conditions -2. Sequential rationality - Each players strategy is optimal whenever she has to move, given her belief and the other players strategies. 2. Consistency of beliefs with strategies Each player s belief is consistent with strategy profile (behavioral strategies of all players) GAME THEORY 29/2

25 Weak Sequential Equilibrium Definition: A weak sequential equilibrium consists of behavioral strategies and beliefs systems satisfying following conditions -2. Sequential rationality - Each players strategy is optimal in the part of the game that follows each of her information sets, given the strategy profile and her belief about the history in the information set that has occurred. In other words for each player i and each information set of player i, according to player s i beliefs, her behavioral strategy gives her the highest possible expected utility so that she has no incentive to deviate from her behavioral strategy in any information set. GAME THEORY 29/2

26 Weak Sequential Equilibrium Definition: A weak sequential equilibrium consists of behavioral strategies and beliefs systems satisfying following conditions Weak consistency of beliefs with strategies For every information set I i reached with positive probability given the players strategies, the probability assigned by the belief system to each history h* in I i is given by P( h * according to strategy profile) h I i P( h according to strategy profile) GAME THEORY 29/2

27 Sequential rationality Each players strategy is optimal whenever she has to move, given her belief and the other players strategies. J(), F 2[⅔] K(), G C(),, 2[⅓] F D() G E() EU(F)=⅔. + ⅓. = ⅓ EU(G)=⅔. + ⅓. = ⅔ G optimal for 2 nd player, given her belief GAME THEORY 29/2

28 Sequential rationality Each players strategy is optimal whenever she has to move, given her belief and the other players strategies. J(), F() 2[⅔] K(), C() G(), F(), 2[⅓] D() G() E() After {C,F} J optimal J() After {ø} E and D optimal E() or D() or {E(p) D(-p)} GAME THEORY 29/2

29 Sequential rationality Each players strategy is optimal whenever she has to move, given her belief and the other players strategies. J(), F() 2[⅔] K(), C() G(), F(), 2[⅓] D() G() E() Player has two optimal strategies DJ, EJ, given player s belief and 2 nd player s strategies GAME THEORY 29/2

30 Consistency of beliefs with strategies Consistency of beliefs with strategies Each player s belief is consistent with strategy profile (behavioral strategies of all players) The idea is that in a steady state, each player s belief must be correct: the probability it assigns to any history must be the probability with which that history occurs if the players adhere to their strategies. If some information set is reached with probability player may have any belief at such information set. GAME THEORY 29/2

31 Consistency of beliefs with strategies The information set is reached with probability zero if P plays E no requirement on beliefs of The 2 nd player J() F 2[⅔] K() G C(),, 2[⅓] F D() G E(),, GAME THEORY 29/2

32 Consistency of beliefs with strategies The information set is reached with probability> C(p) D(q) E(-p-q) J(), 2[p/p+q] F K(), G, 2[q/p+q], F G Requirement 2 nd player belief to C: p/(p+q) Requirement 2 nd player belief to D: q/(p+q) GAME THEORY 29/2

33 Consistency of beliefs with strategies The information set is reached with probability> C(.) D(.6) E(.) J(), F 2[⅓] K(), G, 2[⅔], F G Requirement 2 nd player belief to C:./.9 (.9=.+.6) Requirement 2 nd player belief to D:.6/.9 GAME THEORY 29/2

34 Finding weak sequential equilibria ) If the game has any subgame find at first the weak sequential equilibria of the subgame 2) Identify all possible strategies of all players First way: ) Find all NE of the game in similar way as in the case of dynamic games table with all possible strategies 4) Compute the beliefs consistent with the strategies 5) Check all NE whether they satisfies the 2 conditions for the weak sequential equilibrium GAME THEORY 29/2

35 Finding weak sequential equilibria ) If the game has any subgame find at first the weak sequential equilibria of the subgame 2) Identify all possible strategies of all players SIMPLE GAMES: (2 players, finite number of actions) ) In our quite simple games - start from the beginning by analyzing one after each other strategies of the first player and compute the respective beliefs of the other players, given the strategy of first player 4) Continue by finding the optimal strategies of further players, given their beliefs and strategies of the other players. 5) Check for the equilibrium GAME THEORY 29/2

36 B or S: imperfect information ) No Subgame 2) Strategies of P: B() - bach with probability, S() stravinsky with probability, B(p)S(-p) bach with probability p, stravinsky with -p P P2 P2,,, 2 GAME THEORY 29/2

37 B or S: imperfect information 2) Strategies of P2: B() - bach with probability, S() stravinsky with probability, B(p)S(-p) bach with probability p, stravinsky with -p In the case of perfect information, Strategy defines action for each history Bach, Stravinsky P2 NOW!! Strategy defines action at each Information set, P, P2, 2 GAME THEORY 29/2

38 B or S: imperfect information 2) Strategies of P: B() - bach with probability ) Beliefs of P2 P P2 P2,,, 2 GAME THEORY 29/2

39 B or S: imperfect information 2) Strategies of P: B() - bach with probability ) Beliefs of P2 - belief is consistent with strategy profile P P2[] P2[],,, 2 GAME THEORY 29/2

40 B or S: imperfect information 2) Strategies of P: B() - bach with probability ) Beliefs of P2 4) Optimal strategy of P given her belief and P plays B() P P2[] P2[],,, 2 GAME THEORY 29/2

41 B or S: imperfect information 2) Strategies of P: B() - bach with probability ) Beliefs of P2 4) Optimal strategy of P given her belief and P plays B() Optimal is B() P P2[] P2[],,, 2 GAME THEORY 29/2

42 B or S: imperfect information 2) Strategies of P: B() - bach with probability ) Beliefs of P2 4) Optimal strategy of P given her belief and P plays B() Optimal is B() P 5) Check for equilibrium P2[] P2[],,, 2 GAME THEORY 29/2

43 B or S: imperfect information 2) Strategies of P: B() - bach with probability ) Beliefs of P2 4) Optimal strategy of P given her belief and P plays B() Optimal is B() 5) Check for equilibrium given the strategy of P it is optimal for P to play bach equilibrium OK B() is optimal P2[] P P2[],,, 2 GAME THEORY 29/2

44 B or S: imperfect information 2) Strategies of P: B() - stravinsky with probability ) Beliefs of P2 - belief is consistent with strategy profile P P2[] P2[],,, 2 GAME THEORY 29/2

45 B or S: imperfect information 2) Strategies of P: B() - bach with probability ) Beliefs of P2 4) Optimal strategy of P given her belief and P plays S() Optimal is S() P P2[] P2[],,, 2 GAME THEORY 29/2

46 B or S: imperfect information 2) Strategies of P: B() - bach with probability ) Beliefs of P2 4) Optimal strategy of P given her belief and P plays S() Optimal is S() 5) Check for equilibrium given the strategy of P it is optimal for P to play Stravinsky equilibrium OK S() is optimal P2[] P P2[],,, 2 GAME THEORY 29/2

47 B or S: imperfect information The game has weak sequential equilibria: {P Bach, P2 chooses Bach, P2 believes that history Bach occurs with probability } {P Stravinsky, P2 chooses Stravinsky, PP2 believes that history Stravinsky occurs with probability } There are also other equilibria when P and P2 mix In this simple game Weak sequential eq. coincides with both, SBNE and NE P2,, P2 GAME THEORY 29/2, 2

48 Example ) No Subgame 2) Strategies of P: L() - left with probability, M() middle with probability, R() right with probability, L(p)M(q)R(-p-q) left with probability p, middle with q P RIGHT right with -p-q 4 P2 P2,, 4,, 2 GAME THEORY 29/2

49 Example ) No Subgame 2) Strategies of P2: after {L,M} - L() - left with probability, R() middle with probability, L(p) R(-p) left with probability p, right with -p P2 can recognize whether she is at {R} or {L,M} But she has no choices after {R} P2 P RIGHT P2 4,, 4,, 2 GAME THEORY 29/2

50 Example ) No Subgame 2) Strategies of P: L() - left with probability ) Beliefs of P2 4) Optimal strategy of P2 5) Check for equilibrium Not equilibrium P would choose M() given the P2 strategy. So not the one we assumed in step 2, P2[], P 4, R() P2[] 4, 2 GAME THEORY 29/2

51 Example ) No Subgame 2) Strategies of P: M() - middle with probability ) Beliefs of P2 4) Optimal strategy of P2 5) Check for equilibrium OK equilibrium P would choose M() given the P2 strategy P2[] P R() P2[] 4,, 4,, 2 GAME THEORY 29/2

52 Example ) No Subgame 2) Strategies of P: R() - right with probability ) Beliefs of P2 information set is not reached, so the belief can be arbitrary 4) Optimal strategy of P2 EU(L)=p+(-p)= EU(R)=p+2(-p)=2-p R is never optimal 5) Check for equilibrium NOT equilibrium P would choose M(), P2[p], P 4, R() P2[-p] 4, 2 GAME THEORY 29/2

53 Example ) No Subgame 2) Strategies of P: L(p)M(q)R(-p-q) left with probability p, middle with q right with -p-q ) Beliefs of P2 P 4) Optimal strategy of P2 EU(L)=x+(-x)= EU(R)=x+2(-x)=2-x 5) Check for equilibrium NOT equilibrium P would choose M() given the P2 strategy P2[p/(p+q)],, 4, R(-p-q) 4 P2[q/(p+q)], 2 GAME THEORY 29/2

54 Example The game has only weak sequential equilibrium: {P middle, P2 chooses left after {L,M}, P2 believes that history middle occurs with probability } P RIGHT 4 P2 P2,, 4,, 2 GAME THEORY 29/2

55 Example ) Subgame P plays J C D E J F 2 2 K G,, F G,, GAME THEORY 29/2

56 Example ) Subgame P plays J 2) Strategies of P: J and C(p)D(q)E(-p-q) incorporates all strategiesc(p) ex: p=,q= J E() D(q) E(-p-q) J() F 2[] 2[] K() G,, F G,, GAME THEORY 29/2

57 Example ) Subgame P plays J 2) Strategies of P2: F(p)G(-p) incorporates all strategiesc(p) ex: p= G() D(q) E(-p-q) J() F 2[] 2[] K() G,, F G,, GAME THEORY 29/2

58 Example ) Subgame P plays J 2) Strategies of P: J and E() ) Beliefs of P2 information set is not reached, so the belief can be arbitrary J() F 2[p] K() G C(), 2[-p], F D() G E(),, GAME THEORY 29/2

59 Example ) Subgame P plays J 2) Strategies of P: J and E() ) Beliefs of P2 C() 4) Optimal strategy of P2 J(), F 2[p] K(), G, 2[-p], F D() G E() EU(F)=p. + (-p). = -p EU(G)=p. + (-p). = p p>.5 G, p<.5 F, p=.5 P2 may mixes GAME THEORY 29/2

60 Example ) Subgame P plays J 2) Strategies of P: J and E() ) Beliefs of P2 C() 4) Optimal strategy of P2 5) Check J(), F 2[p] K(), G, 2[-p], F D() G E() p>.5 G, then D and E is optimal for P, So E() is optimal equilibrium OK GAME THEORY 29/2

61 Example ) Subgame P plays J 2) Strategies of P: J and E() ) Beliefs of P2 C() 4) Optimal strategy of P2 5) Check J(), F 2[p] K(), G, 2[-p], F D() G E() p<.5 F, then C for P, So E() is not optimal not equilibrium GAME THEORY 29/2

62 Example ) Subgame P plays J 2) Strategies of P: J and E() ) Beliefs of P2 C() 4) Optimal strategy of P2 5) Check J(), F(x) 2[p] K(), G(-x), 2[-p], F(x) D() G(-x) E() P=.5 F with x, G with -x, then EU(C)= x, EU(D)=2-x, EU(E) = 2 So E() is optimal if x<2/ equilibrium GAME THEORY 29/2

63 Example ) Subgame P plays J 2) Strategies of P: J and C(p)D(q)E(-p-q) ) Belief of P2 C(p) D(q) E(-p-q) J() F 2[] 2[] K() G,, F G,, GAME THEORY 29/2

64 Example ) Subgame P plays J 2) Strategies of P: J and C(p)D(q)E(-p-q) ) Belief of P2 C(p) D(q) E(-p-q) J() 2[p/p+q] F K() G,, F 2[q/p+q] G,, GAME THEORY 29/2

65 Example ) Subgame P plays J 2) Strategies of P: J and C(p)D(q)E(-p-q) ) Belief of P2 C(p) 4) Optimal strategy of P2 D(q) J(), 2[p/p+q] F K(), G,, F 2[q/p+q] G E(-p-q) EU(F)=p/(p+q). + q/(p+q). = q/(p+q) EU(G)=p/(p+q). + q/(p+q). = p/(p+q) p>q G, p<q F, p=q P2 may mixes GAME THEORY 29/2

66 Example ) Subgame P plays J 2) Strategies of P: J and C(p)D(q)E(-p-q) ) Belief of P2 C(p) 4) Optimal strategy of P2 D(q) 5) Check J(), 2[p/p+q] F K(), G,, F 2[q/p+q] G E(-p-q) p>q G P optimal strategy is D() or E() q=>p= Not equilibrium q=p= Not equilibrium GAME THEORY 29/2

67 Example ) Subgame P plays J 2) Strategies of P: J and C(p)D(q)E(-p-q) ) Belief of P2 C(p) 4) Optimal strategy of P2 D(q) 5) Check J(), 2[p/p+q] F K(), G,, F 2[q/p+q] G E(-p-q) p<q F P optimal strategy is C() p=>q= Not equilibrium GAME THEORY 29/2

68 Example ) Subgame P plays J 2) Strategies of P: J and C(p)D(q)E(-p-q) ) Belief of P2 C(p) 4) Optimal strategy of P2 D(q) J(), 2[p/(p+q)] F(x) K(), G(y), F(x), 2[q/(p+q)] G(y) E(-p-q) p=q P2 may mixes (x+y=) P EU: EU(C)=x+y=x, EU(D)=x+2y=+y EU(E)= if x>,y< 2>+y E dominates D player should not play D GAME THEORY 29/2

69 Example ) Subgame P plays J 2) Strategies of P: J and C(p)D(q)E(-p-q) ) Belief of P2 C(p) 4) Optimal strategy of P2 D(q) J(), 2[p/(p+q)] F(x) K(), G(y), F(x), 2[q/(p+q)] G(y) E(-p-q) p=q P2 may mixes (x+y=) E dominates D player should not play D with positive probability p=q=, E() optimal not equilibrium GAME THEORY 29/2

70 Example The game has weak sequential equilibra: {P E and J, P2 chooses F with p<2/, P2 believes that history C occurs C D with probability.5} J, F 2 2 K, G,, F G E {P E and J, P2 chooses G, P2 believes that history C occurs with probability p>.5} GAME THEORY 29/2

71 Summary Dynamic games with incomplete information Weak sequential equilibrium (in our simple cases coincides with perfect Bayesian equilibrium in Gibbons) Gibbons 2.4.A, 4; Osborne NEXT WEEK: weak sequential equilibrium,signaling games GAME THEORY 29/2