Static Games of Incomplete Information Page 2 We can think of the game as beginning with a move by Nature,who to each player assigns a type. Nature's move is imperfectly observed,however:each player observes the type which Nature has bestowed upon her,but no player directly observes the type bestowed upon any other player.We can think of the game which follows as being played by a single type of each player,where at least one player doesn't know which type of some other player she is facing. A Bayesian game Let /=(1,...,n)be the set of players.We refer to a type of player i by 0i,where this type is a member of player i's type space i.e.ii.We denote an n-tuple of types,one for each player-or type profile--byB=(di,,Bn)eo≡XierΘ，whereΘis the type-profile space.When we focus on the types of a player's opponents,we consider deleted type profiles of the form 0-i=(01,,0-1,0+1,,8n)eΘ-i. For our current purposes we will consider the game played after Nature's type assignments as one in strategic form;we are considering static (i.e.simultaneous-move)games of incomplete information.I At a later time we will extend this framework to include dynamic games. The"bigger"game begins when Nature chooses each player's type and reveals it to her alone.In the strategic-form game which follows,each player i ultimately chooses some pure action aiEAi.The n- tuple of actions chosen is the action profile aA=XieAi.2 The payoff player i receives in general depends on the actions a of all players as well as the types e of all players;i.e.ui(a,0),where ui:Ax->R.3 For most of our discussion we will assume that the action and type spaces are finite sets. We denote byi=A(Ai)the space of player-i mixed actions.A typical mixed action for player i is i.A typical deleted mixed-action profile by player i's opponents isXj also =Xiel i Beliefs We assume that there is an objective probability distribution pA()over the type space O,which Nature consults when assigning types.4 In other words,the probability with which Nature draws the type profile =(01,...,0n)-and hence assigns type 01 to player 1,type 02 to player 2,etc.-is p().The 1 See Fudenberg and Tirole [1991:Chapter 6]and/or Gibbons [1992:Chapter 3]for more on static Bayesian games. 2 When studying repeated games we reserved the symbol s to refer to strategies more complicated than stage-game actions,viz.for repeated-game strategies,which were sequences of history-dependent stage-game actions.We do this again,letting a denote actions in the strategic-form game succeeding Nature's revelation of types.and reserving s to refer to a more complicated strategic object,which depends on type,which we'll need in the larger game. It's easy to see why other players'actions should enter into player i's payoff function,but why should other players'types enter into player i's payoffs?Sure,another player's type can influence his action;but this indirect influence of another's type on i's payoff would be captured by the direct effect the other's action has on i's payoff.A reason we should allow in general for a direct dependence of player i's payoff upon others'types is given by the following example.Assume that one firm j has private information about demand, which therefore is captured by his type When firm i competes with firmj.the market outcome and hence firm i's profit will depend on the demand and therefore on firm i's type. 4 For any finite set T we denote by A(T)the set of probability distributions over T. jim@virtualperfection.com Jim Ratliff virtualperfection.com/gametheoryStatic Games of Incomplete Information Page 2 jim@virtualperfection.com Jim Ratliff virtualperfection.com/gametheory We can think of the game as beginning with a move by Nature, who to each player assigns a type. Nature’s move is imperfectly observed, however: each player observes the type which Nature has bestowed upon her, but no player directly observes the type bestowed upon any other player. We can think of the game which follows as being played by a single type of each player, where at least one player doesn’t know which type of some other player she is facing. A Bayesian game Let I={1,…,n} be the set of players. We refer to a type of player i by øi, where this type is a member of player i’s type space Øi; i.e. øi˙Øi. We denote an n-tuple of types, one for each player—or type profile—by ø=(ø1,…,øn)˙ØfiXi˙IØi, where Ø is the type-profile space. When we focus on the types of a player’s opponents, we consider deleted type profiles of the form ø¥i=(ø1,…,øi¥1,øiÁ1,…,øn)˙Ø¥i. For our current purposes we will consider the game played after Nature’s type assignments as one in strategic form; we are considering static (i.e. simultaneous-move) games of incomplete information.1 At a later time we will extend this framework to include dynamic games. The “bigger” game begins when Nature chooses each player’s type and reveals it to her alone. In the strategic-form game which follows, each player i ultimately chooses some pure action ai˙Ai. The n￾tuple of actions chosen is the action profile a˙AfiXi˙IAi.2 The payoff player i receives in general depends on the actions a of all players as well as the types ø of all players; i.e. uiªa,øº, where ui:A˜Ø§Â.3 For most of our discussion we will assume that the action and type spaces are finite sets. We denote by AifiǪAiº the space of player-i mixed actions. A typical mixed action for player i is åi˙Ai. A typical deleted mixed-action profile by player i’s opponents is å¥i˙A¥ifiXj˙I\{i}ÙAj; also AfiXi˙I Ai. Beliefs We assume that there is an objective probability distribution p˙Ǫغ over the type space Ø, which Nature consults when assigning types.4 In other words, the probability with which Nature draws the type profile ø=(ø1,…,øn)—and hence assigns type ø1 to player 1, type ø2 to player 2, etc.—is pªøº. The 1 See Fudenberg and Tirole [1991: Chapter 6] and/or Gibbons [1992: Chapter 3] for more on static Bayesian games. 2 When studying repeated games we reserved the symbol s to refer to strategies more complicated than stage-game actions, viz. for repeated-game strategies, which were sequences of history-dependent stage-game actions. We do this again, letting a denote actions in the strategic-form game succeeding Nature’s revelation of types, and reserving s to refer to a more complicated strategic object, which depends on type, which we’ll need in the larger game. 3 It’s easy to see why other players’ actions should enter into player i’s payoff function, but why should other players’ types enter into player i’s payoffs? Sure, another player’s type can influence his action; but this indirect influence of another’s type on i’s payoff would be captured by the direct effect the other’s action has on i’s payoff. A reason we should allow in general for a direct dependence of player i’s payoff upon others’ types is given by the following example. Assume that one firm j has private information about demand, which therefore is captured by his type øj. When firm i competes with firm j, the market outcome and hence firm i’s profit will depend on the demand and therefore on firm j’s type. 4 For any finite set T we denote by ǪTº the set of probability distributions over T