Leandro Chaves Rêgo. Unawareness in Extensive Form Games. Joint work with: Joseph Halpern (Cornell) Statistics Department, UFPE, Brazil.

Transcription

1 Unawareness in Extensive Form Games Leandro Chaves Rêgo Statistics Department, UFPE, Brazil Joint work with: Joseph Halpern (Cornell) January 2014

2 Motivation Problem: Most work on game theory assumes that: players are aware of all possible moves players are aware of all the other agents who are playing But this isn't always a realistic assumption! e.g., war settings, nancial markets, auctions in large networks,... Goal: Develop new game theoretical models that allow the existence of players who might not be aware of all relevant features of the game.

3 Talk Outline Show how awareness can be modeled in extensive games Show how to generalize standard solution concepts to games with awareness Nash Equilibrium Sequential Equilibrium Remark Although we agree that, in the context of awareness, there might be dierent solution concepts that are more appropriate than generalizations of standard solution concepts, we think it is useful to understand what these generalizations are and why or why not they should be used in the proposed model.

4 Nash Equilibrium The most famous solution concept in game theory is Nash equilibrium. No player changes his strategy even if he knows the strategies of the others. But Nash equilibrium does not always make sense if players are not aware of all moves...

5 A Simple Game A acrossa B acrossb (0,2) downa downb (1,1) (2,3) One Nash equilibrium of this game A plays across A, B plays down B (not unique). But if A is not aware that B can play down B, A will play down A.

6 We need solution concepts that take awareness into account! First step: represent games where players may be unaware We model this by considering a set of extensive games, one describing the objective or true situation and one for each game that at least one of the agents thinks might be the true game in some situation.

7 Representing lack of awareness Γ: an underlying objective extensive form game. Γ describes the moves actually available to players. In our model players awareness are about the histories of Γ. a set of subjective games Γ + based on Γ.

8 Subjective Games A subjective game Γ + based on Γ must satisfy the set of players in Γ + is the same as that of Γ, all moves available to players in Γ + are also available to players in Γ, and the only moves in Γ + that are not in Γ are messages sent by nature that may aect some players' awareness of histories in Γ. these represent players's uncertainty about others' awareness of histories of Γ. we assume these that each move in M + M has the form m = (m i ) i N +, where, intuitively, m i is a message to i that may have the eect of changing i's awareness of histories of Γ.

9 View of the Game We say that i has the same view in histories h and h if i cannot distinguish the actual sequence of moves in h and h, and i receives exactly the same messages regarding awareness in h and h at the same times. In game Γ + based on Γ, h and h are in the same i-information set i player i has the same view in both h and h.

10 Subjective Games Consider the earlier game. Suppose that players A and B are aware of all histories of the game; player A is uncertain as to whether player B is aware of run across A, down B and believes that he is unaware of it with probability p; and the type of player B that is aware of the run across A, down B is aware that player A is aware of all histories, and he knows A is uncertain about B's awareness level and knows the probability p. To represent this, we need three extensive form games.

11 The Objective Game A acrossa B acrossb (0,2) downa downb (1,1) (2,3) Both A and B are aware of all histories of the objective game. But A considers it possible that B is unaware. To represent A's viewpoint, we need another extensive form game.

12 A's view of the game acrossa B.2 acrossb (0,2) unaware (p) downa downb c A.1 (1,1) (2,3) aware (1-p) acrossa B.1 acrossb (0,2) downa downb (1,1) (2,3) At node B.2, B is not aware of the run across A, down B. We need yet another extensive form game to represent this.

13 (A's view of) B's view A.3 across A B.3 across B (0,2) down A (1,1) At node A.3, A is not aware of across A, down B ; neither is B at B.3. Moral: to fully represent a game with awareness we need a set of extensive form games.

14 Game with Awareness A game with awareness based on Γ is a pair Γ = (G, F), where G is a countable set of nite extensive games based on Γ, of which one is Γ; F : (Γ +, h) (Γ h, I ). h is a history in Γ + G; If player i moves at h in Γ + and F(Γ +, h) = (Γ h, I ), then Γ h is the game that i believes to be the true game at h, I is the set of histories in Γ h i considers possible. I is an i-information set histories in I are indistinguishable from i's point of view.

15 Example Continued Γ B the game from an unaware B's viewpoint Γ A representing A's uncertainty about B's awareness downa A.3 (1,1) unaware (p) downa c + A.1 (2,3) (1,1) aware (1-p) downa acrossa B.3 acrossa acrossa downb acrossb B.2 B.1 acrossb acrossb (0,2) (0,2) (0,2) (1,1) (2,3) Γ the objective game A downa B acrossa downb acrossb (0,2) (1,1) (2,3)

16 Example Continued In this game with awareness, we have G = {Γ, Γ A, Γ B }. F(Γ, ) = (Γ A, { aware, unaware }). F(Γ A, unaware,across A ) = (Γ B, { across A }). F(Γ A, aware,across A ) = (Γ, { across A }).

17 Consistency Conditions F satises conditions capturing our intuition regarding awareness and ensuring that players remember the game they considered possible and the moves they made. Assume F(Γ +, h) = (Γ h, I ). Thus, C1. h I i h is a history in H h where i has the same view as in h. C2. Every actual sequence of moves in Γ h must also be in Γ +. C3. If player i has the same view in any two histories of any two subjective games, then i must believe in the same game and has the same information at both of these histories. C4. A player never forgets some history he was previously aware of and he changes the game that he considers possible only if he becomes aware of more moves of the objective game. C5. There exists some plausible history in I, where a plausible history h is one such that every player that moved in h is aware of the chosen move. although players may be aware of histories that are not plausible, these can never be played.

18 Adding Awareness to Games By C3, the game that a player considers possible is determined by the histories he is aware of, and how he became aware of them. This is relevant because player i's beliefs about the game may well be aected by how player j strategically made i aware of various moves E.g., if player 1 is initially unaware that he can move left and he can become aware of it by either hearing it from player 2 or player 3, then his beliefs about the true game, including his beliefs about what other players are aware of, may depend on who tells player 1 about this possibility.

19 Canonical Representation A standard extensive game Γ can be identied with the game ({Γ}, F), where F(Γ, h) = (Γ, I ) for h I. This is the canonical representation of Γ as a game with awareness. Intuition: In the canonical representation of Γ as a game with awareness, all players understand the structure of the objective game Γ and this is common knowledge among players. A standard game can be viewed as a special case of a game with awareness, where the objective game is common knowledge.

20 Strategies In a standard game, a behavioral strategy for player i is a function from i-information sets to a distribution over moves. i must do the same thing at histories i cannot tell apart. A strategy is a universal plan, describing what i will do in every possible circumstance. In games with awareness, this does not make sense! A player cannot plan in advance what he will do when he becomes aware of new moves.

21 Local Strategies In a game Γ = (G, F) with awareness, we consider a collection of local strategies, one for each game an agent may consider to be the true one in some situation. Intuitively, a local strategy σ i,γ for game Γ is the strategy that i would use if i were called upon to play and i thought that the true game was Γ. There may be no relationship between the strategies σ i,γ games Γ. for dierent Essentially, if Γ 1 Γ 2, we treat the version of player i who believes Γ 1 to be the true game, as a dierent player from the version of player i who believes Γ 2 to be the true game, since discovering about a new move may lead i to totally revise his strategy.

22 Aside: Information Sets In a game with awareness, if F(Γ +, h) = (Γ h, I ) and P + (h) = i, then there are 3 candidates for the information set that contains h: The information set in Γ + that contains h, but i may not be aware of all the histories in such information set; From the point of view of a player who believes the game is Γ +, these are the histories that i cannot distinguish from h; The information set I in the game that i believes, Γ h ; and These are the histories considered possible by i while moving at h; The set of all histories in all subjective games where i has the same view as in h, we call this third one a generalized information set. By C3, i has the same beliefs about the game in all such histories. Thus, a local strategy must choose the same move in all such histories.

23 Notation Dene Pr σ ( h) to be the conditional probability distribution induced by σ over the possible histories of the game given that the current history is h. Intuitively, Pr σ (h h) is 0 if h is not a prex of h, 1 if h = h, and the product of the probability of each of the moves in the path from h to h if h is a prex of h. We denote Pr σ (h ) by Pr σ (h ).

24 Generalized Nash Equilibrium Intuition: σ is a generalized Nash equilibrium if for every player i, if i believes he is playing game Γ, then his local strategy σ i,γ is a best response to the local strategies of other players in Γ. The local strategies of the other players are part of σ. Formally, σ is a generalized Nash equilibrium of a game Γ = (G, F) with awareness if, for every player i, game Γ G i, and local strategy σ for i in Γ, EU i,γ ( σ ) EU i,γ (( σ (i,γ ), σ)), where EU i,γ ( σ ) = z Z Pr σ (z)u i (z).

25 Major Results σ is a Nash equilibrium of a standard game Γ i σ is a (generalized) Nash equilibrium of the canonical representation of Γ as a game with awareness. Theorem: Every game with awareness has at least one generalized Nash equilibrium.

26 Sketch of the Proof We associate with a game Γ with awareness a standard game Γ ν and show that σ is a (generalized) Nash equilibrium of Γ i σ is a Nash equilibrium of Γ ν. Given a game Γ = (G, F) with awareness, let ν be a probability on G that assigns each game in G positive probability. Γ ν essentially glues together all the games Γ G. Non plausible moves are removed before we glue the games together in Γ ν. Nature makes an initial move in Γ ν, choosing game Γ with probability ν(γ ). The set of players in Γ ν is (i, Γ ). Player (i, Γ ) has utility zero for all runs of Γ ν whose rst move is not Γ.

27 Γ ν (A, Γ B ) acrossa acrossb (0,0,0,2) downa (B, Γ B ) (0,0,1,1) Γ B acrossa acrossb (0,0,0,0) unaware (p) downa c Γ A c (1,0,0,0) aware (1-p) (A, Γ A ) acrossa acrossb (0,0,0,0) Γ downa downb (1,0,0,0) (B, Γ) (2,0,0,0) acrossa acrossb (0,2,0,0) downa downb (0,1,0,0) (0,3,0,0)

28 Subtleties 1 Γ Γ 1 1 L R L R L R L R R R (20,20) (5,10) (4,10) (10,5) (5,10) (10,5) 1 Γ 2 L R 2 2 L R L R (20,20) This game has two generalized Nash equilibria in pure strategies. (5,10) (4,10) In both, player 1 chooses R in Γ 1 and L in Γ 2 and player 2 chooses L at the left node of Γ 2. Finally, in the rst generalized Nash equilibrium, player 2 chooses L at the right node of Γ 2, while in the second, he chooses R at such node. (10,5)

29 Subtleties In both equilibria, player 2 has false beliefs about whether he is on the equilibrium path in Γ. At the left node, he believes that he is on the equilibrium path, while he is not. Even though his choice at this node does not aect his objective equilibrium payo, he believes it does. At the right node, although he is on the equilibrium path, he does not believe it. Although his choice at this node aects his objective equilibrium payo, he believes it does not, and, in the second equilibria, he plays a dominated action at this node.

30 Subtleties One way to rationalize it, would be to think that players choose their strategies at the beginning of the game. But such interpretation would be problematic if we modify Γ by adding a dummy move for player 2 at the beginning of the game, where player 2 at this initial moment believed the game to be Γ 1. In that case, it would not make sense to think about player 2 choosing his local strategy for Γ 2 at the beginning of the game, as he is not aware of it at this point. Thus, it is possible to reach an information set on the equilibrium path in Γ, where player 2 moves, but player 2 believes that he is at an information set in a dierent game, one to which he assigns probability 0, and thus there are no constraints on what he must do there in a generalized Nash equilibrium. Moral: We need better solution concepts!

31 Other Solution Concepts There exist a number of variants of Nash equilibrium proposed in the literature, such as perfect equilibrium, proper equilibrium, sequential equilibrium, and rationalizability. Using our framework, we can generalize these solution concepts: Idea: treat player i who considers Γ 1 to be the true game as dierent from player i who considers Γ 2 to be the true game. Dierent versions of player i can use dierent strategies. Each one best responds (in the sense appropriate for that solution concept), given his view of the game. Showing that every game has an equilibrium of the appropriate type can be nontrivial.

32 Sequential Equilibrium Sequential equilibrium is dened with respect to an assessment, a pair ( σ, µ) where σ is a strategy prole, and µ is a belief system, i.e., a function that determines for each information set I a probability µ I over the histories in I.

33 Belief System Intuitively, if I is an i-information set, µ I is i's subjective assessment of the relative likelihood of the histories in I. Roughly speaking, a belief system is consistent with a strategy prole σ, if when I is reached with positive probability according to σ, then µ I is obtained by conditioning; and when I is reached with probability 0, then it is obtained by a limit of a sequence of alternative belief systems consistent with a sequence of perturbations of the original strategy prole such that each available move is played with strictly positive probability. The standard interpretation is that the player who moves at I, must believe some other player(s) made a small mistake while implementing σ.

34 Sequential Equilibrium An assessment is a sequential equilibrium if for all players i, at every i-information set, (a) i chooses a best response given his beliefs and the strategies of other players, and (b) i's beliefs are consistent with the strategy prole being played. In the simple game, the unique sequential equilibrium is (across A,down B ). A across A B across B (0,1) down A down B (1,3) (3,2)

35 Generalizing Solution Concepts As in the case of Nash equilibrium, sequential equilibrium is not a reasonable solution concept for games with awareness. Players cannot know about strategies of other players that involve moves that they are not aware of A player cannot plan in advance what he will do when he becomes aware of new moves. Generalized sequential equilibrium requires a generalized notion of assessment

36 Generalized Assessment Let (Γ h, I ) = {(Γ, h) : F(Γ, h) = (Γ h, I )} represent a generalized information set. Nodes where player thinks he is playing game Γ h and is in information set I. A generalized belief system µ determines for each (Γ, I ) a probability µ Γ,I over the set {(Γ, h) : h I } This support consists of the histories that i considers possible while moving in one of the histories in (Γ, I ). In many cases, this support is a proper subset of (Γ, I ). A generalized assessment is a pair ( σ, µ), where σ is a generalized strategy prole and µ is a generalized belief system.

37 Generalized Sequential Equilibrium Formally, a generalized assessment ( σ, µ ) is a generalized sequential equilibrium of a game Γ = (G, F) with awareness if the following conditions are satised: Generalized sequential rationality. For every player i, generalized i-information set (Γ, I ), and local strategy σ for i in Γ, EU i,γ (( σ, µ ) I ) EU i,γ ((( σ (i,γ ), σ), µ ) I ), where EU i,γ (( σ, µ ) I ) = h I z Z µ Γ,I (Γ, h) Pr σ (z h)u i (z).

38 Generalized Sequential Equilibrium Consistency between generalized belief system and generalized strategy prole. If, for every generalized information set (Γ, I ), h I Pr σ (h) > 0, then for all h I µ Γ,I (Γ, h) = Pr σ (h) h I Pr σ (h ). Otherwise, there exists a sequence of generalized assessments ( σ i, µ i ) such that every player chooses all of his moves, that he is aware of, with positive probability, µ i is consistent with σ i and lim i ( σ i, µ i ) = ( σ, µ ).

39 Generalized Sequential Equilibrium Intuition: ( σ, µ ) is a generalized sequential equilibrium of a game with awareness if, for every player i, if i believes he is at information set I of game Γ, then his local strategy σ i,γ is a best response to the local strategies of other players in Γ given his beliefs about the histories in I, as given by µ (Γ,I ) ; and µ is consistent with σ. It is not dicult to see that ( σ, µ ) is a sequential equilibrium of a standard game Γ i ( σ, µ ) is a (generalized) sequential equilibrium of the canonical representation of Γ as a game with awareness.

40 Another Example (1,1) A 1.1 Consider this game L R 2.1 (-10,-1) l r l r (2,-2) (-1,-2) (0,-1)

41 Another Example Suppose that both players 1 and 2 are aware of all runs of the game, but player 1 (falsely) believes that player 2 is aware only of the runs not involving L and believes that player 1 is aware of these runs as well. player 2 is aware of all of this; that is, player 2's view of the game is the same as the objective game Γ shown in the previous gure.

42 Another Example While moving at node 1.1, player 1 considers the true game to be identical to the modeler's game except that from player 1's point of view, while moving at 2.1, player 2 believes the true game is Γ 2.2, shown below. (1,1) A 1.2 R 2.2 l r (-1,-2) (0,-1)

43 Gen. Seq. Equilibrium This game has a unique generalized sequential equilibrium where: player 2 chooses r and player 1 chooses A in Γ 2.2 player 1 chooses L at node 1.1 player 2 chooses l at node 2.1 in Γ m. Thus, if players follow their equilibrium strategies, the payo vector is ( 10, 1). In this situation, player 2 would be better o is she could let player 1 know that she is aware of move L, since then player 1 would play A and both players would receive 1. On the other hand, if we slightly modify the game by making u 2 ( L, l ) = 3, then player 2 would benet from the fact that 1 believes that she is unaware of move L.

44 Existence Theorem: Every game with awareness has at least one generalized sequential equilibrium. Sketch of Proof: To prove existence of generalized Nash equilibrium in a game Γ with awareness, we constructed a standard game Γ ν such that σ is a (generalized) NE of Γ i σ is a NE of Γ ν.

45 Sketch of Proof (cont'd) Unfortunately, the 1-1 correspondence between the Nash equilibria of Γ ν and the generalized Nash equilibria of Γ breaks down for sequential equilibria. In a standard game, all histories in I are in the support of the belief system µ I. In a game with awareness, not all histories in (Γ, I ) are in the support of µ (Γ,I ). the agent is not aware of all these histories!

46 Sketch of Proof (cont'd) We consider conditional sequential equilibrium with respect to K, a generalization of sequential equilibrium for standard games. For each information set I, K(I ) I is the support of µ I. Players are certain that the histories in I K(I ) do not occur. We modify the standard proof of existence of sequential equilibrium to show that, for all (acceptable) K, every standard game has a conditional sequential equilibrium with respect to K. We show, for an appropriate K, there is a 1-1 correspondence between the conditional sequential equilibria of Γ ν w.r.t. K and the generalized sequential equilibria of Γ.

47 Awareness of Unawareness Sometimes players may be aware that they are unaware of relevant moves: War settings: you know that an enemy may have new technologies of which you are not aware Delaying a decision Chess you may become aware of new issues tomorrow lack of awareness" inability to compute"

48 Modeling Awareness of Unawareness If i is aware that j can make a move at h that i is not aware of, then j can make a virtual move at h in i's subjective representation of the game The payos after a virtual move reect i's beliefs about the outcome after the move. Just like computers associate a value to a board position in chess The subjective game representation according to player i may include some virtual moves to him; i may believe that he will become aware of other moves in the future. Again, there is guaranteed to be a generalized sequential equilibrium.

49 Unexpected Moves Consider the following game: (3,3) e g (5,5) u 2 f 1 h (0,0) 1 m f 1 i (3,2) d e j (4,0) (3,3) (2,0)

50 Unexpected Moves Suppose that player 2 is unaware of the move u for player 1 and in addition believes that this is commonly believed to be the actual game. Suppose player 1 knows all this and therefore believes he is playing the objective game m f i (3,2) d e j (4,0) (3,3) (2,0) In the unique generalized sequential equilibrium in pure strategies, player 1 chooses the strategy dgi in the objective game Γ and the strategy di in game Γ 2, and player 2 chooses the strategy e in game Γ 2.

51 Unexpected Moves In this equilibrium, when player 2 moves, she believes that player 1 made a mistake by following his equilibrium strategy d. Remark: One could try to argue that in an awareness context when player 2 sees player 1 doing an unexpected move, he could become aware of something or at least become aware that he is unaware of something that player 1 is. We argue that since in our model awareness is given exogenously, if that unexpected move changes player 2 awareness of histories, then this should be already described in the game that represents player 2's view of the game (by allowing for awareness of unawareness). For us, generalized sequential equilibrium makes as much sense in a game with awareness as sequential equilibrium does in standard extensive form games. Although, we agree that in some other model, where awareness is not exogenously given, another solution concept would make more sense.

52 Unexpected Moves Suppose instead player 2 believes that, at his second information set, player 1 has some move available that she is unaware of, but that would lead them to a better payo m f v i (a,b) (3,2) d e j (4,0) (3,3) (2,0)

53 Unexpected Moves If a > 4 and b > 3, then there is a unique generalized sequential equilibrium, where player 1 chooses the strategy ugi in the objective game Γ and the strategy mv in game Γ 2v, and player 2 chooses the strategy f in game Γ 2v. Note that player 2 does not think player 1 has made a mistake. Now she believes that 1 has some available move that she is not aware of, but that she believes end up given both of them a better payo. It is worth mentioning that in this equilibrium player 2 falsely believes that player 1 chooses m at the beginning of the game; in fact, player 1 chooses u.

54 Lack of Common knowledge Our model is exible enough to deal with lack of common knowledge about the structure of the underlying game even if (lack of) awareness is not involved. This lack of common knowledge can be about who moves in a node of the game tree, information sets, consequences of strategies. The key idea is that we remove the assumption that all extensive games are based on the same objective game Γ. Also in this case, under reasonable assumptions on F, we can prove that for nite games there exists a generalized sequential equilibrium.

55 Some Other Results In the model, we presented here, even though the function F was deterministic we were able to model uncertainty about unawareness, introducing additional moves of nature in the subjective games. If we want to model static or normal form games with awareness, we need to make some modications in our model since in normal form games there are no moves of nature. The essential dierence is to make the function F to be probabilistic and to add a vector in the denition of a normal form game representing the players' view about the objective game in some situation. Some care must be taken to take into account the possibility of correlation between the games believed by the players.

56 Some Other Results (cont'd) A methodology similar to the one presented here can be applied to model a (cooperative) 2-Player Bargaining Problem with Unawareness. The idea is to consider a set of augmented bargaining problems representing the players' view about the objective bargaining situation. An augmented bargaining problem is a standard bargaining problem added with a vector of players' views about the objective bargaining situation. A generalization of Nash's Bargaining Solution can be proved to exist and satisfy axioms similar to the axioms in the standard Nash's axiomatic model.

57 Some Open Problems We assumed perfect recall, what are the issues involved when we allow imperfect recall and unawareness? Are there ecient computational techniques for computing generalized Sequential equilibrium? The set G of augmented games can be quite large, or even innite. Are there conditions under which higher-order awareness becomes less important? Key unanswered question: what solution concepts are most appropriate in games with awareness?

58 References 1 Halpern, J. Y. ; Rêgo, L. C.. Extensive games with possibly unaware players. Mathematical Social Sciences, forthcoming. 2 Rêgo, Leandro C. ; Halpern, J. Y.. Generalized solution concepts in games with possibly unaware players. International Journal of Game Theory, v.41, p , 2012.