Advanced Micro 1 Lecture 14: Dynamic Games Equilibrium Concepts

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1 Advanced Micro 1 Lecture 14: Dynamic Games quilibrium Concepts Nicolas Schutz Nicolas Schutz Dynamic Games: quilibrium Concepts 1 / 79

2 Plan 1 Nash equilibrium and the normal form 2 Subgame-perfect equilibrium and backward induction 3 Weak perfect Bayesian equilibrium 4 Sequential equilibrium Nicolas Schutz Dynamic Games: quilibrium Concepts 2 / 79

3 Plan 1 Nash equilibrium and the normal form 2 Subgame-perfect equilibrium and backward induction 3 Weak perfect Bayesian equilibrium 4 Sequential equilibrium Nicolas Schutz Dynamic Games: quilibrium Concepts 3 / 79

4 Fix a finite extensive game with perfect recall Γ = ( N, H, P, f, (I i ) i N, ( g i )i N). Remember that S i, the set of pure strategies of player i, is the set of functions and let S = j N S i. s i : I i I i s i (I i ) A(I i ), Remember also that the expected payoff of player i given pure strategy profile s S is: u i (s) = ω(s)(h)g i (h). Definition 1 h Z The normal form (or strategic form) of extensive game Γ is game G = (N, (S i ) i N, (u i ) i N ). A pure strategy profile s S is a pure-strategy Nash equilibrium of Γ if it is a pure-strategy Nash equilibrium of G. Nicolas Schutz Dynamic Games: quilibrium Concepts 4 / 79

5 Remember that Σ i is the set of mixed strategies of Γ. Since Σ i = (S i ), it is also the set of mixed strategies of G. As usual, let Σ = j N Σ i. Σ is the set of mixed strategy profiles of Γ (and G). The expected payoff (in Γ) of player i when mixed strategy profile σ Σ is played is: ũ i (σ) = ω(σ)(h)g i (h), = = = h Z σ(s)ω(s)(h) g i(h), h Z s S σ(s) ω(s)(h)g i (h), s S h Z σ(s)u i (s), s S which is also the payoff (in G s mixed extension G) of player i when mixed strategy profile σ Σ is played. Nicolas Schutz Dynamic Games: quilibrium Concepts 5 / 79

6 Definition 2 A mixed strategy profile σ Σ is a Nash equilibrium in mixed strategies (or, simply, a Nash equilibrium) of extensive game Γ if it is a mixed-strategy Nash equilibrium of normal form game G. Nicolas Schutz Dynamic Games: quilibrium Concepts 6 / 79

7 Some examples: Player 1 The normal form associated with this extensive form is: C (3,3) C K Player 2 K C (1,4) (4,1) K (0,0) C K C 3, 3 1, 4 K 4, 1 0, 0 This normal form game has three Nash equilibria ((C, K), (K, C), ( 1 2 K C, 1 2 K + 1 2C)). These strategy profiles are the three Nash equilibria of the extensive form game. Nicolas Schutz Dynamic Games: quilibrium Concepts 7 / 79

8 Nature fficient, prob. p Incumbent Inefficient, prob. 1 p Incumbent B NB B NB ntrant N N N N ( ) ( ) ( ) 2 1 ( ) 3 0 ( ) 0 1 ( ) 2 0 ( ) 2 1 ( ) 3 0 Nicolas Schutz Dynamic Games: quilibrium Concepts 8 / 79

9 Let s look for the normal form of this game. The entrant has only one information set, so only two possible pure strategies: and N. The incumbent has two information sets, so four possible pure strategies: (B, B) (i.e., build in information set (ε) and build in information set (ι) ). (B, NB), (NB, B), (NB, NB). xpected payoffs are easy to calculate: If the incumbent plays (NB, B) and the entrant plays, then the outcome of this strategy profile is: p (ε, NB, ) + (1 p) (ι, B, ). This yields expected payoffs of p2+(1 p)0 for the incumbent and p1+(1 p)( 1) for the entrant. Nicolas Schutz Dynamic Games: quilibrium Concepts 9 / 79

10 So the normal form of this simple entry game can be summarized as follows: N (B,B) (1.5p, -1) (2+1.5p, 0) (B,NB) (2-0.5p, 1-2p) (3+0.5p, 0) (NB,B) (2p, 2p-1) (2-p, 0) (NB,NB) (2, 1) (3,0) We already know what are the pure-strategy Nash equilibria of this game: (NB, NB),, and (B, NB), N (if p 1/2). These are the only pure-strategy Nash equilibria of the extensive game. Two approaches to deal with incomplete information: Construct the expanded game and look for its Nash equilibria. Construct an extensive game (replace incomplete information by nature s moves), write it in normal form, and look for the Nash equilibria of the normal form. These two approaches are equivalent. Nicolas Schutz Dynamic Games: quilibrium Concepts 10 / 79

11 xample: The pilot and terrorist game Pilot NYC Cuba Terrorist Terrorist Bomb Don t bomb Bomb Don t bomb -1, -1 2, 0-1, -1 1, 1 Pure strategies? For the pilot, two pure strategies: S 1 = {NYC, Cuba}. For the terrorist, four pure strategies: S 2 = {(B, B), (B, NB), (NB, B), (NB, NB)} (where (NB, B) means don t bomb in NYC, but bomb in Cuba ). Nicolas Schutz Dynamic Games: quilibrium Concepts 11 / 79

12 Here is the normal form of the pilot-terrorist game: (B,B) (B,NB) (NB,B) (NB,NB) NYC -1, -1-1, -1 2, 0 2, 0 Cuba -1, -1 1, 1-1, -1 1, 1 liminate (B, B), which is strictly dominated by (NB, NB): (B,NB) (NB,B) (NB,NB) NYC -1, -1 2, 0 2, 0 Cuba 1, 1-1, -1 1, 1 Three Nash equilibria in pure strategies: (NYC, (NB, NB)), (NYC, (NB, B)), (Cuba, (B, NB)). What about mixed strategy equilibria? Nicolas Schutz Dynamic Games: quilibrium Concepts 12 / 79

13 quilibria in which the pilot mixes? If the pilot puts > 0 weight on NYC and Cuba, then the terrorist s best response is (NB, NB). But if the terrorist plays (NB, NB), then the pilot does not want to mix. So there is no Nash equilibrium in which the pilot mixes. quilibria in which the terrorist mixes? Assume first that the pilot plays NYC with probability 1: Terrorist is indifferent between (NB, NB) and (NB, B), and he clearly does not want to play (B, NB). Suppose the terrorist plays p(nb, NB) + (1 p)(nb, B). Then, the pilot s best response is indeed to play NYC (for any p). This gives us a continuum of mixed-strategy equilibria. Nicolas Schutz Dynamic Games: quilibrium Concepts 13 / 79

14 Now assume that the pilot plays Cuba with probability 1: Terrorist is indifferent between (NB, NB) and (B, NB). Suppose the terrorist plays p(nb, NB) + (1 p)(b, NB). If the pilot plays Cuba, then he gets a payoff of 1 (terrorist never bombs in Cuba). If he plays NYC, then he gets a payoff of p2 + (1 p)( 1) = 3p 1. The pilot is fine with playing Cuba provided that p 2/3. This gives us again a continuum of mixed-strategy equilibria. To sum up, the Nash equilibria of the pilot and terrorist game are: ( NYC, p(nb, NB) + (1 p)(nb, B) ), where p [0, 1]. ( Cuba, p(nb, NB) + (1 p)(b, NB) ), where p [ 2 3, 1]. (pure strategy equilibria are just special cases) Nicolas Schutz Dynamic Games: quilibrium Concepts 14 / 79

15 Plan 1 Nash equilibrium and the normal form 2 Subgame-perfect equilibrium and backward induction 3 Weak perfect Bayesian equilibrium 4 Sequential equilibrium Nicolas Schutz Dynamic Games: quilibrium Concepts 15 / 79

16 So we get a lot of Nash equilibria in the pilot and terrorist game. We now argue that most of these equilibria (actually: all but one) have undesirable properties. Consider equilibrium Cuba, (B, NB). It is supported by the following reasoning: Pilot: I m facing a robot which plays B in NYC and NB in Cuba. Clearly, I should fly to Cuba. Terrorist: I m facing a robot which flies to Cuba. I m indifferent between (NB, NB) and (B, NB), since we will never end up in NYC. The implicit assumption here (which is a direct consequence of our use of the normal form) is that the terrorist s robot can perfectly commit to playing B in NYC. But this is not how the game is really played: the pilot first flies to a destination, then the terrorist decides whether to bomb the plane. Nicolas Schutz Dynamic Games: quilibrium Concepts 16 / 79

17 Here is a simple reasoning for the pilot which (a) exploits the dynamic / sequential nature of the game, (b) eliminates equilibrium Cuba, (B, NB): I m facing the terrorist s robot. This robot says I ll bomb in NYC but not in Cuba. But what if I did fly to NYC? If the terrorist lets his robot play, then he ll get -1. But he can also ditch his robot, not bomb the plane, and get 0. I know the terrorist is rational, so, if I do fly to NYC, then he ll ditch his robot. I ll get a payoff of 2 > 1. So I ll fly to NYC. What this reasoning says is that: Threat I ll bomb the plane if you fly to NYC is not a credible threat. Strategy (B, NB) is not sequentially rational. In fact, in this game, the only sequentially rational (mixed) strategy is (NB, NB). Nicolas Schutz Dynamic Games: quilibrium Concepts 17 / 79

18 [Another example: Nash equilibria in the Cournot-Stackelberg game.] Firm 1 N Firm 2 N N Firm 1 T S 5, 0 0, 5 0, 0 Firm 2 T S T S -2, -2-1, -1-1, -1 2, 2 Nicolas Schutz Dynamic Games: quilibrium Concepts 18 / 79

19 This is just a two-stage game of observed actions, with the following timing: 1 Firms 1 and 2 simultaneously decide whether to enter. 2 If both firms enter, then firms 1 and 2 simultaneously decide whether to play Tough or Soft. We won t provide a complete characterization of the Nash equilibria of this game. Let s focus on the following three pure-strategy Nash equilibria: (, S), (, S), (, T), (N, S), (N, S), (, T). quilibrium (, T), (N, S) (and its twin brother) looks suspicious: The reason why firm 2 does not enter is that firm 1 threatens firm 2 to play T in case of entry. But what if firm 2 did enter. Then, firm 2 will play a simple simultaneous moves game with firm 1. Firm 2 should expect to play a Nash equilibrium with firm 1. The only Nash equilibrium in this part of the game tree is (S, S). Nicolas Schutz Dynamic Games: quilibrium Concepts 19 / 79

20 Again, the threat of playing T in case entry takes place is not credible. Player 1 should expect player 2 to play optimally (and vice versa) after entry takes place. We capture this idea as follows: Players 1 and 2 should expect a Nash equilibrium (in the market competition subgame) to emerge after entry has taken place. There is a unique Nash equilibrium in this subgame: (S, S). It follows that the unique profile of strategies consistent with sequential rationality is ((, S), (, S)). Now let s extend these ideas to general extensive games. Nicolas Schutz Dynamic Games: quilibrium Concepts 20 / 79

21 Notation: Take a history h = (a k ) 1 k K H, and L K. Let h = (a k ) 1 k L ( H) and h = (a k ) L+1 k K. We write h = (h, h ). Nicolas Schutz Dynamic Games: quilibrium Concepts 21 / 79

22 Definition 3 Let h H\Z such that: I P(h) (h) is a singleton, For every h such that (h, h ) H\Z, for every h H\Z, if h I P((h,h ))((h, h )), then there exists ĥ such that h = (h, ĥ). The ( subgame that follows history h is the extensive game Γ(h) = N, H h, P h, f h, (I i h ) i N, ( h ) ) g i, where H h = {h : (h, h ) H}. i N For every h H h \ Z h, P h (h ) = P((h, h )). For every h H h \ Z h such that P h (h ) = 0, f h (. h ) = f (. (h, h )). For all i N, write I i = { } I j i j J i. Let J i = { j J i : h s.t. (h, h ) I j i}. For all j J, let Ij = { h : (h, h ) I j i i i}. I i h = { } I j is the information partition of player i in subgame Γ(h). i j J i For all i N, h Z h, g i h (h ) = g i ((h, h )). Nicolas Schutz Dynamic Games: quilibrium Concepts 22 / 79

23 Notice that the entire game is always a subgame of itself: Γ( ) = Γ. Subgames are sometimes called proper subgames. xamples of subgames: The pilot and terrorist game has exactly three subgames: The whole game: Pilot NYC Cuba Terrorist Terrorist Bomb Don t bomb Bomb Don t bomb -1, -1 2, 0-1, -1 1, 1 Nicolas Schutz Dynamic Games: quilibrium Concepts 23 / 79

24 The NYC subgame : Terrorist The Cuba subgame : Terrorist Bomb Don t bomb Bomb Don t bomb -1, -1 2, 0-1, -1 1, 1 Nicolas Schutz Dynamic Games: quilibrium Concepts 24 / 79

25 Things that are not subgames in the pilot-terrorist game: Pilot NYC Cuba Terrorist Bomb Bomb Terrorist Don t bomb -1, -1-1, -1 1, 1 Nicolas Schutz Dynamic Games: quilibrium Concepts 25 / 79

26 The entry game (with ex post competition) has exactly two subgames: The game itself: Firm 1 N Firm 2 N N Firm 1 T S 5, 0 0, 5 0, 0 Firm 2 T S T S -2, -2-1, -1-1, -1 2, 2 Nicolas Schutz Dynamic Games: quilibrium Concepts 26 / 79

27 And the competition subgame: Firm 1 T S Firm 2 T S T S -2, -2-1, -1-1, -1 2, 2 Nicolas Schutz Dynamic Games: quilibrium Concepts 27 / 79

28 Things that are not subgames: Firm 2 N Firm 1 T Firm 2 S 5, 0 T S T S -2, -2-1, -1-1, -1 2, 2 We can t chop up firm 2 s information set! (because this implicitly changes the information structure of the game.) Nicolas Schutz Dynamic Games: quilibrium Concepts 28 / 79

29 Now, look at the entry game with inefficient/efficient incumbent: This game has exactly one subgame: the whole game Nature fficient, prob. p Incumbent Inefficient, prob. 1 p Incumbent B NB B NB ntrant N N N N ( ) ( ) ( ) 2 1 ( ) 3 0 ( ) 0 1 ( ) 2 0 ( ) 2 1 ( ) 3 0 Nicolas Schutz Dynamic Games: quilibrium Concepts 29 / 79

30 The following is not a subgame: Incumbent B NB ntrant N N ( ) ( ) ( ) 2 1 ( ) 3 0 Here, we re chopping up the entrant s information set. Nicolas Schutz Dynamic Games: quilibrium Concepts 30 / 79

31 Let Σ b i h be the set of behavior strategies of player i in subgame Γ(h). Remember that any behavior strategy can be viewed as a mixed strategy, and vice versa. Definition 4 Let Γ(h) be a subgame of Γ, i N, and σ i Σ b i a behavior strategy of player i in game Γ. For all h H h such that P h (h ) = i, we let σ i h (I i h (h )) = σ i (I i ((h, h ))). σ i h Σ b i h is the restriction of strategy σ i to subgame Γ(h). We say that strategy σ i induces strategy σ i h in subgame Γ(h). Let Σ b h = j N Σ b j be the set of behavior strategy profiles in subgame Γ(h). h Similar notation for pure / mixed strategies. Nicolas Schutz Dynamic Games: quilibrium Concepts 31 / 79

32 Selten s subgame perfect equilibrium: Definition 5 Behavior strategy profile σ Σ b is a subgame-perfect equilibrium (or subgameperfect Nash equilibrium) if it induces a Nash equilibrium in every subgame. Formally: for every subgame Γ(h), for every i N, for every σ i Σb i h, g i h (σ i h, σ i h ) g i h ( σ i, σ i h ). xample: Fix a profile of behavior strategies in the pilot / terrorist game: Pilot: σ P [0, 1] is the probability that the pilot goes to Cuba. Terrorist: σ T = (σ NYC, σ Cuba ) [0, 1] 2, where σ j is the probability that the T T T terrorist does not bomb the plane in location j. Consider the Cuba subgame : The restriction of the terrorist s strategy in this subgame is σ Cuba (the pilot T does nothing in this subgame). There is a unique profile of strategies consistent with Nash equilibrium in this subgame: σ Cuba = 1 (never bomb the plane). T Nicolas Schutz Dynamic Games: quilibrium Concepts 32 / 79

33 By the same token, there is unique profile of strategies consistent with Nash equilibrium in the NYC subgame: σ NYC T = 1. So we re left with a unique behavior strategy profile for the terrorist: (1, 1). Now look at subgame Γ( ) (the whole game). Since the terrorist plays (1, 1), the pilot s unique best response is to play σ P = 0 (go to NYC with probability 1). Clearly, if the pilot plays NYC with probability 1, then (1, 1) is a best response for the terrorist. Conclusion: Strategy profile NYC, (NB, NB) is the only subgame-perfect equilibrium of the pilot-terrorist game. Remark: If strategy profile σ is a subgame-perfect equilibrium (SP), then it is also a Nash equilibrium. As can be seen above, some Nash equilibria are not subgame-perfect. SP is a refinement of Nash. Nicolas Schutz Dynamic Games: quilibrium Concepts 33 / 79

34 Proposition 1 Let Γ(h) be a subgame of Γ. Suppose that strategy profile σ s is an SP of Γ(h). Let ˆΓ be the reduced game obtained by replacing subgame Γ(h) by a terminal history with payoffs equal to those arising from play of σ s. Then: (i) Let σ Σ such that σ h = σ s, and suppose σ is an SP of Γ. Then, the restriction of σ to ˆΓ constitutes an SP of ˆΓ. (ii) Let ˆσ be an SP of ˆΓ. Define σ Σ as follows: for all i N, for all I i I i, { σ s σ i (I i ) = i (I i) if I i I i h, ˆσ i (I i ) otherwise. Then, σ is an SP of Γ. Nicolas Schutz Dynamic Games: quilibrium Concepts 34 / 79

35 This suggests the following backward induction procedure: 1 Identify the Nash equilibria for each of the final subgames (i.e., those that have no strict subgames). 2 Select one Nash equilibrium in each of these final subgames. Derive the reduced extensive form game in which these final subgames are replaced by the payoffs that result in these subgames when players use these equilibrium strategies. 3 Repeat steps 1 and 2. Continue the procedure until every move in Γ is determined. This collection of moves at the various information sets of Γ constitutes a profile of SP strategies. Remark: Multiplicity. Nicolas Schutz Dynamic Games: quilibrium Concepts 35 / 79

36 xamples: Pilot NYC Cuba Terrorist Terrorist Bomb Don t bomb Bomb Don t bomb -1, -1 2, 0-1, -1 1, 1 Pilot This tells us, again, that the only SP is NYC, (NB, NB). 2, 0 1, 1 Nicolas Schutz Dynamic Games: quilibrium Concepts 36 / 79

37 xamples (Cont d) Firm 1 N Firm 2 N N Firm 1 T S 5, 0 0, 5 0, 0 Firm 2 T S T S -2, -2-1, -1-1, -1 2, 2 Nicolas Schutz Dynamic Games: quilibrium Concepts 37 / 79

38 Firm 1 N N Firm 2 N 2, 2 5, 0 0, 5 0, 0 So the unique SP is (, S), (, S). Nicolas Schutz Dynamic Games: quilibrium Concepts 38 / 79

39 xamples (Cont d): What if there are multiple equilibria in a subgame? Firm 1 N Firm 2 N N Firm 1 T S 5, 0 0, 5 0, 0 Firm 2 T S T S -1, -1 0, -2-2, 0 2, 2 Nicolas Schutz Dynamic Games: quilibrium Concepts 39 / 79

40 The normal form game played in the competition subgame is: Three Nash equilibria: S T S 2, 2-2, 0 T 0, -2-1, -1 1 (S, S), with equilibrium payoffs (2, 2), 2 (T, T), with equilibrium payoffs (-1, -1), 3 ( 1 3 S T, 1 3 S T), with equilibrium payoff ( 2 3, ) 2 3. Start by selecting the first one. Nicolas Schutz Dynamic Games: quilibrium Concepts 40 / 79

41 Firm 1 N Firm 2 N N 2, 2 5, 0 0, 5 0, 0 Therefore, (, S), (, S) is an SP. Nicolas Schutz Dynamic Games: quilibrium Concepts 41 / 79

42 Now, select the second Nash equilibrium: Firm 1 N Firm 2 N N -1, -1 5, 0 0, 5 0, 0 Write this in normal form: N -1, -1 5, 0 N 0, 5 0, 0 Nicolas Schutz Dynamic Games: quilibrium Concepts 42 / 79

43 Three Nash equilibria:, N, N,, ( N, N) This gives us three additional SPs: (, T), (N, T), (N, T), (, T), ( N, T), ( N, T). Nicolas Schutz Dynamic Games: quilibrium Concepts 43 / 79

44 Finally, let s select the last Nash equilibrium: Firm 1 N Firm 2 N N 2 3, 2 3 5, 0 0, 5 0, 0 Write this in normal form: N 2 3, 2 3 5, 0 N 0, 5 0, 0 Nicolas Schutz Dynamic Games: quilibrium Concepts 44 / 79

45 Again, three Nash equilibria:, N, N,, ( N, N) This gives us three additional SPs: (, 1 3 S T), ( N, 1 3 S T), ( N, 1 3 S T), (, 1 3 S T), ( N, 1 3 S T), ( N, 1 3 S T). Nicolas Schutz Dynamic Games: quilibrium Concepts 45 / 79

46 So the game has exactly 7 SPs: (, S), (, S), (, T), (N, T), (N, T), (, T), ( N, T), ( N, T). (, 1 3 S T), ( N, 1 3 S T), ( N, 1 3 S T), (, 1 3 S T), ( N, 1 3 S T), ( N, 1 3 S T). Nicolas Schutz Dynamic Games: quilibrium Concepts 46 / 79

47 On path vs. off path: Fix a behavior strategy profile σ Σ b. Remember that, if h H, then ω(σ)(h) is the probability that history h will be encountered at some point during the play of the game. Now assume that profile σ is a Nash equilibrium. We say that subgame Γ(h) is on the equilibrium path if ω(σ)(h) > 0. Subgame Γ(h) is off the equilibrium path if ω(σ)(h) = 0. If σ is a Nash equilibrium and if ω(σ)(h) > 0, then σ h is a Nash equilibrium of subgame Γ(h). Put differently, players behaviors should be optimal in every subgame on the equilibrium path. The Nash equilibrium concept does not put any optimality constraints on subgames that are off the equilibrium path. Nicolas Schutz Dynamic Games: quilibrium Concepts 47 / 79

48 This is what allowed us to sustain strategy profile Cuba, (B, NB) as a Nash equilibrium: The NYC subgame is off the equilibrium path, so the terrorist can choose whatever crazy strategy he wants there. Remember how we proved that there is no equilibrium in which the pilot mixes. If the pilot puts positive weights on Cuba and NYC, then everything is on path. This forces the terrorist to behave optimally everywhere. But then, the pilot does not want to mix anymore. The difference between Nash equilibrium and SP is that SP imposes optimality in all subgames, including those that are off the equilibrium path. One last remark: If σ is a Nash equilibrium such that every subgame is on the equilibrium path (i.e., if ω(σ)(h) > 0 for every subgame Γ(h)), then σ is a SP. This happens, for instance, if σ is completely mixed (i.e., for every player i, for every I i, σ(i i ) assigns positive probabilities to every element of A(I i )). Nicolas Schutz Dynamic Games: quilibrium Concepts 48 / 79

49 A particular class of games: Multistage games of observed actions. Here is an informal description: Set of players N. T stages. In stage t (t {1, 2,..., T}), a subset of players (say: N t N) take actions simultaneously. At the beginning of each stage, all players observe all the actions which were taken before. Usually, nature doesn t move. (Alternatively: assume that nature moves at the beginning of some stages. Nature s move in period t may not be common knowledge in the current period. What matters is that nature s moves in previous stages is observed by everyone). We shouldn t use the Nash equilibrium concept to solve these games, because lots of equilibria would rely on non-credible threats (except if T = 1... ). It is perfectly fine to use subgame-perfect equilibrium to solve these games. (In particular, in these games, the sets of subgame-perfect equilibria and sequential equilibria coincide) Nicolas Schutz Dynamic Games: quilibrium Concepts 49 / 79

50 Some examples of multistage games of observed actions: Strategic trade policy: Three countries: A, B and C. Firm A is located in country A, firm B is located in country B. Both firms sell in country C. Governments can subsidize exports and maximize (domestic) social welfare. Firms set quantities and maximize profit. Timing: 1 Governments A and B set their subsidies simultaneously. 2 Subsidies become common knowledge. Firms A and B set their quantities simultaneously. Competition in R&D: N firms: i = 1,..., N. ach firm chooses its own price and its own R&D intensity. Firms maximize profit. Timing: 1 Firms choose their R&D intensities simultaneously. 2 R&D intensities become common knowledge. Firms their prices simultaneously. Nicolas Schutz Dynamic Games: quilibrium Concepts 50 / 79

51 Plan 1 Nash equilibrium and the normal form 2 Subgame-perfect equilibrium and backward induction 3 Weak perfect Bayesian equilibrium 4 Sequential equilibrium Nicolas Schutz Dynamic Games: quilibrium Concepts 51 / 79

52 Player 1 L M R 2, 2 l r Player 2 l r 3, 1 0, 0 0, 2 1, 1 The following strategy profiles are Nash equilibria: (M, l), (L, r), ( L, pl + (1 p)r ) (for all 0 p 2/3). Since the game does not have any subgames (except the whole game itself), all these equilibria are also subgame-perfect. Nicolas Schutz Dynamic Games: quilibrium Concepts 52 / 79

53 quilibrium (L, r) and the continuum of mixed-strategy equilibria look very suspicious. They rely on the following threat from player 2: If you don t play L, then I ll punish you by playing r with a high probability. This does not look like a credible threat: conditional on player 2 s information set being reached, player 2 is worse off playing r than playing l, no matter which action player 1 took. This non-credible threat is not ruled out by subgame-perfection, because of the lack of subgames in this game. Nicolas Schutz Dynamic Games: quilibrium Concepts 53 / 79

54 Now let s think about this problem in more detail: Suppose player 2 gets the move. He knows that he s in information set {(L), (M)}, but he doesn t know whether past history is (L) or (M). As usual in economics, let s replace he doesn t know by he believes that the history is L with probability µ [0, 1] and R with probability 1 µ. If he plays l, then his expected payoff is µ + 2(1 µ). If he plays r, then his expected payoff is (1 µ). Therefore, player 2 wants to play l (and this is true no matter what his beliefs are). Anticipating this, player 1 should choose M. Bottom line: Forcing players to have beliefs allows us to impose sequential rationality at decisions nodes where subgame-perfection has no bite. The next step is to generalize this. Nicolas Schutz Dynamic Games: quilibrium Concepts 54 / 79

55 Let I = i N I i be the set of all information sets of Γ. Definition 6 A belief system is a function µ : I I µ(i) (I). In information set I, player P(I) believes that the history is h( I) with probability µ(i)(h). Let σ be a profile of behavior strategies. Player P(I) should use his beliefs to construct a probability distribution over terminal nodes (which will allow him to maximize his expected payoff). Denote this probability distribution by ω(σ, µ I). Let h = (a k ) 1 k K Z and, for every L K, let h L = (a k ) 1 k L. Nicolas Schutz Dynamic Games: quilibrium Concepts 55 / 79

56 Distinguish two cases: 1 Assume there is no L K such that h L I. Then, ω(σ, µ I)(h) = 0. 2 Assume there exists L K such that h L I. Notice that L is unique, by perfect recall. Then, (where we let σ 0 = f (..)). Definition 7 K 1 ω(σ, µ I)(h) = µ(i)(h L ) k=l σ P(h k ) ω(σ, µ I) is called the outcome of (σ, µ) conditional on I. ( IP(h k )(h k ) ) (a k+1 ), For a given belief system µ and profile of strategies σ, we can also define the expected payoff of player i = P(I) conditional on being in information set I: ũ i (σ µ, I) = ω(σ, µ I)(h)g i (h). h Z Nicolas Schutz Dynamic Games: quilibrium Concepts 56 / 79

57 Here is a formal definition of sequential rationality: Definition 8 Strategy profile σ Σ is sequentially rational at information set I given belief system µ if, denoting i = P(I), ( σ i Σ i) (ũi ((σ i, σ i ) µ, I) ũ i ((σ i, σ i) µ, I) ). If σ satisfies this condition for all information sets, then we say that σ is sequentially rational given belief system µ. Next, we impose some consistency requirements on the belief system. The idea is that players should use equilibrium strategy profile σ to form beliefs about the information set they are in. Start with the easy case: Assume σ is completely mixed. Formally: for every I I, for every a A(I), σ P(I) (a) > 0. Nicolas Schutz Dynamic Games: quilibrium Concepts 57 / 79

58 Let I I and h = (a 1,..., a K ) I. Use the usual h L notation. We know that the probability that history h will be met during the play of the game is: K 1 ( ω(σ)(h) = IP(h )(h k ) ) (a k+1 ), k k=0 which is > 0, since all the σ s are > 0. σ P(h k ) [As usual, we redefine σ 0 as f. We assume that nature plays every a with strictly positive probability. If there exists h and a A(h) such that f (a h) = 0, then history (h, a) and all its successors are never reached, no matter what σ is. If this is the case, then we can define an equivalent game without history (h, a) and all its successors.] The probability that information set I will be met during the play of the game is: ω(σ)(i) = ω(σ)(h) > 0. h I Nicolas Schutz Dynamic Games: quilibrium Concepts 58 / 79

59 The probability that the current history is h I conditional on being in information set I can then be derived using Bayes rule: ω(σ I)(h) = ω(σ)(h) ω(σ)(i) = ω(σ)(h) h I ω(σ)(h ) µ(i)(h). Clearly, µ(i)(h) > 0 for all h I, and h I µ(i)(h) = 1, so µ is a belief system. So when σ is completely mixed, there is a simple and intuitive way of deriving a belief system: Apply Bayes rule. When σ is not completely mixed, some information sets may be reached with probability 0: There may exist I I such that ω(σ)(i) = 0 (in this case, we also have that ω(σ)(h) = 0 for all h I). Bayes rule cannot be applied in these information sets. Nicolas Schutz Dynamic Games: quilibrium Concepts 59 / 79

60 The literature proposes various ways of dealing with consistency of belief systems in information sets where Bayes rule cannot be applied. Let s start with the weakest consistency requirement: Definition 9 Belief system µ is weakly consistent with strategy profile σ if it is derived from σ through Bayes rule whenever possible. Formally, if I I, h I and ω(σ)(i) > 0, then: µ(i)(h) = ω(σ)(h) ω(σ)(i). Definition 10 A profile of strategies and belief system (σ, µ) is a weak perfect Bayesian equilibrium (weak PB) if it has the following properties: (i) Strategy profile σ is sequentially rational given belief system µ. (ii) Belief system µ is weakly consistent with strategy profile σ. Beliefs are part of the equilibrium concept. Nicolas Schutz Dynamic Games: quilibrium Concepts 60 / 79

61 Why is this the weakest consistency requirement? Remember that Nash equilibrium is correct conjectures + rationality. xample: Consider Chicken. Suppose players coordinate on the (C, K) Nash equilibrium. Then, player 2 expects player 1 to play C with probability 1. So, in his information set, player 2 believes that the history is (C) with probability 1. He will act optimally given these beliefs. So if we throw away Bayesian updating whenever possible, then we essentially throw away correct conjectures, which means that we also throw away the Nash equilibrium concept. Fix a weak PB, (σ, µ). We say that information set I I is on the equilibrium path if ω(σ)(i) > 0. Weak PB imposes Bayes rule in information sets that are on the equilibrium path. The concept does not put any restrictions on beliefs in information sets which are off the equilibrium path. We will see later on why this can be problematic. Nicolas Schutz Dynamic Games: quilibrium Concepts 61 / 79

62 More on the relationship between weak PB and Nash: Proposition 2 A strategy profile σ is a Nash equilibrium if and only if there exists a belief system µ such that: (i) Strategy profile σ is sequentially rational given belief system µ at all information sets I I such that ω(σ)(i) > 0. (ii) Belief system µ is weakly consistent with strategy profile σ. So Nash only imposes sequential rationality on the equilibrium path. Let (σ, µ) be a weak PB. Then, (i) and (ii) hold. Therefore, σ is a Nash equilibrium. Weak PB is a refinement of Nash. Nicolas Schutz Dynamic Games: quilibrium Concepts 62 / 79

63 Suppose strategy profile σ is a completely mixed Nash equilibrium. Then, a unique belief system is consistent with Bayes rule whenever possible (i.e., always). Call this belief system µ. All information sets are on path. By (i), σ is then sequentially rational given µ at all information sets. Therefore, (σ, µ) is a weak PB. So if a strategy profile is a completely mixed Nash equilibrium, then this strategy profile can be sustained in a weak PB. Nicolas Schutz Dynamic Games: quilibrium Concepts 63 / 79

64 Player 1 L M R 2, 2 l r Player 2 l r 3, 1 0, 0 0, 2 1, 1 Take 0 p 2/3 and look at Nash equilibrium σ = ( L, pl + (1 p)r ). Let ρ [0, 1], and suppose players beliefs are: µ({ })( ) = 1, µ({(m), (R)})((M)) = ρ and µ({(m), (R)})((R)) = 1 ρ. Clearly, ω(σ)({(m), (R)}) = 0, so belief system µ is weakly consistent. σ is sequentially rational given µ at the only information set which is on path (information set { }). Therefore, σ is indeed a Nash equilibrium. Nicolas Schutz Dynamic Games: quilibrium Concepts 64 / 79

65 Weak PB also imposes sequential rationality in information sets off the equilibrium path. As we discussed earlier, this implies that none of these Nash equilibria are weak PBs. We are left with a unique candidate: (M, l). We have already shown that (M, l), µ is a weak PB for any belief system µ. Nicolas Schutz Dynamic Games: quilibrium Concepts 65 / 79

66 Problems with the weak PB concept: Player 1 2,2,2 A C B Player 2 D Here is a weak PB of this game: Player 1 plays A, player 2 plays C, player 3 plays F, all with probability 1. Players 1 and 2 s beliefs are trivial. Player 3 F F If player 3 s information set is reached, then he believes that player 2 played D with probability 1. 4,4,4 1,1,3 5,3,1 4,0,2 These beliefs are weakly consistent (since player 3 s information set is off the equilibrium path). ach player is acting sequentially rationally given his beliefs. Nicolas Schutz Dynamic Games: quilibrium Concepts 66 / 79

67 Problems with the weak PB concept: Player 1 A B 2,2,2 Player 2 C D Player 3 But this game has a unique subgameperfect equilibrium: (A, C, )! So weak PB does not imply SP. (And we already know that SP does not imply weak PB). F F 4,4,4 1,1,3 5,3,1 4,0,2 Notice that player 3 s beliefs are really silly: he believes that player 2 will play a strictly dominated strategy (conditional on player 2 s information set being reached) with probability 1. This is not ruled out by weak PB. Nicolas Schutz Dynamic Games: quilibrium Concepts 67 / 79

68 Nature L, prob. 0.5 R, prob. 0.5 Player 1 O I I O 1,1 l r Player 2 l r 1,1 2, 2 1, 0 2, 2 0, 3 Here is a weak PB: Player 1 plays O and player 2 plays r. If his information set is reached, then player 2 believes that the history is (R, I) with probability 1. (Player 1 s beliefs are pinned down by Bayes rule: (L) with probability 1/2 and (R) with probability 1/2) Nicolas Schutz Dynamic Games: quilibrium Concepts 68 / 79

69 Player 2 s beliefs are unreasonable: Player 1 is signalling what he doesn t know. Since player 1 does not know what nature did, player 2 shouldn t be able to infer anything about what nature did from player 1 s choice of action. Here, the only reasonable belief for player 2 is the original prior: L (resp. R) with probability 1/2. Another way to see this: player 2 s beliefs are not structurally consistent: There is no (mixed or behavior strategy) of player 1, σ 1, such that player 2 s beliefs can be derived through Bayes rule. A (completely mixed) strategy for player 1 is just a number σ 1 (0, 1). Then: P((L, I)) = P(L)P(I) = 1 σ = P(R)P(I) = P((R, I)). 2 Therefore, P((L, I) {(L, I), (R, I)}) = 1 2. The only structurally consistent belief for player 2 is, again, the original prior. Nicolas Schutz Dynamic Games: quilibrium Concepts 69 / 79

70 Bottom line: Weak PB does not impose any restrictions on beliefs in information sets off the equilibrium path. This sometimes leads to very unreasonable equilibria. There are two answers to this in the literature: Restrict attention to a special class of games: Bayesian extensive games with observed actions, and define a new concept: Perfect Bayesian equilibrium (PB). Among other things, PB applies Bayes rule in some information sets, even though they are off the equilibrium path (example), contains a no signalling what you don t know condition. PB tends to be used in the applied theory literature. Define a sequential equilibrium. Sequential equilibrium tends to be used in the game theory literature. Nicolas Schutz Dynamic Games: quilibrium Concepts 70 / 79

71 Plan 1 Nash equilibrium and the normal form 2 Subgame-perfect equilibrium and backward induction 3 Weak perfect Bayesian equilibrium 4 Sequential equilibrium Nicolas Schutz Dynamic Games: quilibrium Concepts 71 / 79

72 Definition 11 Belief system µ is consistent with strategy profile σ if there exists a sequence of completely mixed behavior strategies (σ n ) n N such that the following two conditions are satisfied: (i) σ n n σ. (ii) For every n N, for every I I and h I, let Then, µ n n µ. µ n (I)(h) = ω(σn )(h) ω(σ n )(I). Of course, consistency implies weak consistency. Nicolas Schutz Dynamic Games: quilibrium Concepts 72 / 79

73 Kreps and Wilson s sequential equilibrium: Definition 12 A profile of strategies and belief system (σ, µ) is a sequential equilibrium if it has the following properties: (i) Strategy profile σ is sequentially rational given belief system µ. (ii) Belief system µ is consistent with strategy profile σ. [In the sequential equilibrium literature, a strategy profile / belief system pair (σ, µ) is called an assessment.] Proposition 3 If (σ, µ) is a sequential equilibrium, then it is a weak PB. Nicolas Schutz Dynamic Games: quilibrium Concepts 73 / 79

74 Let s apply the sequential equilibrium concept to this game: Player 1 A B 2,2,2 C Player 3 Player 2 D F F 4,4,4 1,1,3 5,3,1 4,0,2 Nicolas Schutz Dynamic Games: quilibrium Concepts 74 / 79

75 Fix an assessment (σ, µ): Let σ = (σ 1, σ 2, σ 3 ) [0, 1] 3. σ 1 is the probability that player 1 plays A, σ 2 is the probability that player 2 plays C, and σ 3 is the probability that player 3 plays. Let µ [0, 1]: in his information set, player 3 believes that player 2 has played C with probability µ. Suppose (σ, µ) is a sequential equilibrium. Then, there exists a sequence of completely mixed strategies (σ n ) n N such that σ n n σ, and the corresponding belief systems µ n n µ. Let n N. The probability of getting history (B, C) is (1 σ n 1 )σn 2. The probability of getting history (B, D) is (1 σ n 1 )(1 σn 2 ). Applying Bayes rule, we get that µ n = σ 2 n. Taking limits, it follows that µ = σ 2. Nicolas Schutz Dynamic Games: quilibrium Concepts 75 / 79

76 So if player 3 s information set is reached, then he believes that player 2 has played C with probability σ 2. Sequential rationality (of player 3 conditional on µ = σ 2 ) implies that player 3 should play a best response to σ 2 (conditional on his information set being reached). Sequential rationality of player 2 implies that σ 2 should be a best response to σ 3 (conditional on player 2 s information set being reached). Therefore, (σ 2, σ 3 ) should be a Nash equilibrium in the subgame which starts when player 2 has the move. The only Nash equilibrium in this subgame is (C, ). It follows that σ 2 = σ 3 = 1 in any sequential equilibrium. Clearly, player 1 s best response to this is σ 1 = 0 (i.e., play B with probability 1). Conclusion: the only sequential equilibrium of this game is σ = (B, C, ), µ = 1. [Need one more step] Nicolas Schutz Dynamic Games: quilibrium Concepts 76 / 79

77 We have just shown, in this particular game, that any sequential equilibrium is subgame-perfect. Here is a general result: Proposition 4 If (σ, µ) is a sequential equilibrium, then σ is a subgame-perfect equilibrium. Let s summarize what we know about the equilibrium concepts we ve seen: Sequential equilibrium subgame-perfect equilibrium Nash equilibrium. Sequential equilibrium Weak perfect Bayesian equilibrium Nash equilibrium. Some weak PBs are not subgame-perfect, some SPs are not weak PBs. If σ is completely mixed, then sequential equilibrium SP weak PB Nash equilibrium. One more result: Theorem 1 very finite extensive game has a sequential equilibrium. Nicolas Schutz Dynamic Games: quilibrium Concepts 77 / 79

78 Nature L, prob. 0.5 R, prob. 0.5 Player 1 O I I O 1,1 l r Player 2 l r 1,1 2, 2 1, 0 2, 2 0, 3 Suppose that (σ, µ) is a sequential equilibrium. Take a sequence (σ n ) n N such that (σ n ) σ. n Apply Bayes rule and take the limit to show that µ = 1 2 that player 2 assigns to L in his information set). (where µ is the belief Nicolas Schutz Dynamic Games: quilibrium Concepts 78 / 79

79 Sequential rationality for player 2 (given µ = 1/2) implies that he should play l with probability 1 if his information set is reached. Anticipating this, player 1 plays I with probability 1. Conclusion: The game has a unique sequential equilibrium: σ = (I, l), Player 1 s beliefs are pinned down by Bayes rule. Player 2 s beliefs: µ = 1 2. Nicolas Schutz Dynamic Games: quilibrium Concepts 79 / 79

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