Расширенная форма представления игр в теории игр

Wide ranging representation of a game in game theory

In game theory, an extensive form game is a specification of a game allowing (as the name suggests) for the explicit representation of a number of key aspects, like the sequencing of players' possible moves, their choices at every decision point, the (possibly imperfect) information each player has about the other player's moves when they make a decision, and their payoffs for all possible game outcomes. Extensive form games also allow for the representation of incomplete information in the form of chance events modeled as "moves by nature". Extensive form representations differ from normal form in that they provide a more complete description of the game in question, whereas normal form simply boils down the game into a payoff matrix.

It may be the case that a player does not know exactly what the payoffs of the game are or of what type their opponents are. This sort of game has incomplete information. In extensive form it is represented as a game with complete but imperfect information using the so called Harsanyi transformation. This transformation introduces to the game the notion of nature's choice or God's choice. Consider a game consisting of an employer considering whether to hire a job applicant. The job applicant's ability might be one of two things: high or low. Their ability level is random; they either have low ability with probability 1/3 or high ability with probability 2/3. In this case, it is convenient to model nature as another player of sorts who chooses the applicant's ability according to those probabilities. Nature however does not have any payoffs. Nature's choice is represented in the game tree by a non filled node. Edges coming from a nature's choice node are labelled with the probability of the event it represents occurring. The game on the left is one of complete information (all the players and payoffs are known to everyone) but of imperfect information (the employer doesn't know what nature's move was.) The initial node is in the centre and it is not filled, so nature moves first. Nature selects with the same probability the type of player 1 (which in this game is tantamount to selecting the payoffs in the subgame played), either t1 or t2. Player 1 has distinct information sets for these; i. e. player 1 knows what type they are (this need not be the case). However, player 2 does not observe nature's choice. They do not know the type of player 1; however, in this game they do observe player 1's actions; i. e. there is perfect information. Indeed, it is now appropriate to alter the above definition of complete information: at every stage in the game, every player knows what has been played by the other players. In the case of private information, every player knows what has been played by nature. Information sets are represented as before by broken lines. In this game, if nature selects t1 as player 1's type, the game played will be like the very first game described, except that player 2 does not know it (and the very fact that this cuts through their information sets disqualify it from subgame status). There is one separating perfect Bayesian equilibrium; i. e. an equilibrium in which different types do different things. If both types play the same action (pooling), an equilibrium cannot be sustained. If both play D, player 2 can only form the belief that they are on either node in the information set with probability 1/2 (because this is the chance of seeing either type). Player 2 maximises their payoff by playing D' However, if they play D' , type 2 would prefer to play U. This cannot be an equilibrium. If both types play U, player 2 again forms the belief that they are at either node with probability 1/2. In this case player 2 plays D' , but then type 1 prefers to play D.

If type 1 plays U and type 2 plays D, player 2 will play D' whatever action they observe, but then type 1 prefers D. The only equilibrium hence is with type 1 playing D, type 2 playing U and player 2 playing U' if they observe D and randomising if they observe U. Through their actions, player 1 has signalled their type to player 2.

It may be that a player has an infinite number of possible actions to choose from at a particular decision node. The device used to represent this is an arc joining two edges protruding from the decision node in question. If the action space is a continuum between two numbers, the lower and upper delimiting numbers are placed at the bottom and top of the arc respectively, usually with a variable that is used to express the payoffs. The infinite number of decision nodes that could result are represented by a single node placed in the centre of the arc. A similar device is used to represent action spaces that, whilst not infinite, are large enough to prove impractical to represent with an edge for each action. The tree on the left represents such a game, either with infinite action spaces (any real number between 0 and 5000) or with very large action spaces (perhaps any integer between 0 and 5000). This would be specified elsewhere. Here, it will be supposed that it is the former and, for concreteness, it will be supposed it represents two firms engaged in Stackelberg competition. The payoffs to the firms are represented on the left, with q 1 and q 2 as the strategy they adopt and c 1 and c 2 as some constants (here marginal costs to each firm). The subgame perfect Nash equilibria of this game can be found by taking the first partial derivative of each payoff function with respect to the follower's (firm 2) strategy variable (q 2) and finding its best response function, The same process can be done for the leader except that in calculating its profit, it knows that firm 2 will play the above response and so this can be substituted into its maximisation problem. It can then solve for q 1 by taking the first derivative, yielding Feeding this into firm 2's best response function, and is the subgame perfect Nash equilibrium.