Extensive form game

An extensive form game is a specification of a game in game theory. This form represents the game as a tree. Each node (called a decision node) represents every possible state of play of the game as it is played. Play begins at a unique initial node, and flows through the tree along a path determined by the players until a terminal node is reached, where play ends and payoffs are assigned to all players. Each non-terminal node belongs to a player; that player chooses among the possible moves at that node, each possible move is an edge leading from that node to another node.

The extensive form is an alternative to the normal form representation. Unlike the normal form, the extensive form allows explicit modelling of interactions in which a player makes more than one move during the game, and moves contingent upon varying states.

Representation
A complete extensive form representation specifies:
 * 1) the players of a game
 * 2) for every player every opportunity they have to move
 * 3) what each player can do at each of their moves
 * 4) what each player knows for every move
 * 5) the payoffs received by every player for every possible combination of moves.



The game on the right has two players: 1 and 2. The numbers by every non-terminal node indicate to which player that decision node belongs. The numbers by every terminal node represent the payoffs to the players (e.g. 2,1 represents a payoff of 2 to player 1 and a payoff of 1 to player 2). The labels by every edge of the graph are the name of the action that that edge represents.

The initial node belongs to player 1, indicating that that player moves first. Play according to the tree is as follows: player 1 chooses between U and D; player 2 observes player 1's choice and then chooses between U'  and D' . The payoffs are as specified in the tree. There are four outcomes represented by the four terminal nodes of the tree: (U,U'), (U,D'), (D,U') and (D,D'). The payoffs associated with each outcome respectively are as follows (0,0), (2,1), (1,2) and (3,1).

If player 1 plays D, player 2 will play U'  to maximise his payoff and so player 1 will only receive 1. However, if player 1 plays U, player 2 maximises his payoff by playing D'  and player 1 receives 2. Player 1 prefers 2 to 1 and so will play U and player 2 will play D' . This is the subgame perfect equilibrium.

Infinite Action Space
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 protuding 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 latter 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 q1 and q2 as the strategy they adopt and c1 and c2 as some constants (here marginal costs to each firm). The 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 (q2) and finding its best response function, $$q2(q1)=(5000-q1-c2)/2$$. 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 q1 by taking the first derivative, yielding $$q1*=(5000+c2-2c1)/2$$. Feeding this into firm 2's best response function, $$q2*=(5000+2c1-3c2)/4$$ and (q1*,q2*) is the Nash equilibrium. For example, if c1=c2=1000, the Nash equilibrium is (2000, 1000).

Imperfect Information
An advantage of representing the game in this way is that it is clear what the order of play is. The tree shows clearly that player 1 moves first and player 2 observes this move. However, in some games play does not occur like this. One player does not always observe the choice of another (for example, moves may be simultaneous or a move may be hidden). An information set is a set of decision nodes such that:
 * 1) Every node in the set belongs to one player.
 * 2) When play reaches the information set, the player with the move cannot differentiate between nodes within the information set, i.e. if the information set contains more than one node, the player to whom that set belongs does not know which node in the set has been reached.

In extensive form, an information set is indicated by a dotted line connecting all nodes in that set or sometimes by a loop drawn around all the nodes in that set.



If a game has an information set with more than one member that game is said to have imperfect information. A game with perfect information is such that at any stage of the game, every player knows exactly what has taken place earlier in the game, i.e. every information set is a singleton set. Any game without perfect information has imperfect information.

The game on the left is the same as the above game except that player 2 does not know what player 1 does when he comes to play. The first game described has perfect information; the game on the left does not. If both players are rational and both know that both players are rational and everything that is known by any player is known to be known by every player (i.e. player 1 knows player 2 knows that player 1 is rational and player 2 knows this, etc. ad infinitum), play in the first game will be as follows: player 1 knows that if he plays U, player 2 will play D'  (because for player 2 a payoff of 1 is preferable to a payoff of 0) and so player 1 will receive 2. However, if player 1 plays D, player 2 will play U'  (because to player 2 a payoff of 2 is better than a payoff of 1) and player 1 will receive 1. Hence, in the first game, the equilibrium will be (U, D' ) because player 1 prefers to receive 2 to 1 and so will play U and so player 2 will play D' .

In the second game it is less clear: player 2 cannot observe player 1's move. Player 1 would like to fool player 2 into thinking he has played U when he has actually played D so that player 2 will play D'  and player 1 will receive 3. In fact in the second game there is a perfect Bayesian equilibrium where player 1 plays D and player 2 plays U'  and player 2 holds the belief that player will definitely play D. In this equilibrium, every strategy is rational given the beliefs held and every belief is consistent with the strategies played. Notice how the imperfection of information changes the outcome of the game.

In games with infinite action spaces and imperfect information, non-singleton information sets are represented, if necessary, by inserting a dotted line connecting the (non-nodal) endpoints behind the arc described above or by dashing the arc itself. In the Stackelberg game described above, if the second player had not observed the first player's move the game would no longer fit the Stackelberg model; it would be Cournot competition.

Incomplete Information
It may be the case that a player does not know exactly what the payoffs of the game are or of what type his opponents are. This sort of game has incomplete information. It is represented in extensive form by introducing 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. His ability level is random; he is low ability with probability 1/3 and 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. This representation is the result of Harsanyi transformation.



The game on the left has incomplete information. 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 he is (this need not be the case). However, player 2 does not observe nature's choice. He does not know the type of player 1; however, in this game he does observe player 1's actions, i.e. there is perfect information. Indeed, it is now appropriate to alter the above definition of perfect information: at every stage in the game, every player knows what has been played by the other players. In the case of complete 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 his 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 he is on either node in the information set with probability 1/2 (because this is the chance of seeing either type). Player 2 maximises his payoff by playing D' . However, if he plays 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 he is 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 he observes, 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 he observes D and randomising if he observes U. Through his actions, player 1 has signalled his type to player 2.

Axiomatic Formulation
Game theory is known to be a mathematical theory. It is possible to carry out an axiomatic formulation of the above-stated game tree structure.

Formally, a finite game in extensive form is a structure $$ \Gamma = <\mathcal{K}, \mathbf{H}, [ (\mathbf{H}_i)_{i \in \mathcal{I} } ], \{ A(H) \}_{H \in \mathbf{H} } ], a, \rho, u >$$ where:
 * $$\mathcal{K} = $$ is a finite tree with a set of nodes $$ V $$, a unique initial node $$v^0 \in V$$, a set of terminal nodes $$T \subset V$$ (let $$D = V \setminus T$$ be a set of decision nodes) and an immediate predecessor function $$ p: V \rightarrow D $$ on which the rules of the game are represented,
 * $$\mathbf{H}$$ is a partition of $$D$$ called an information partition,
 * $$ A(H) $$ is a set of actions available for each information set $$H \in \mathbf{H}$$ which forms a partition on the set of all actions $$\mathcal{A}$$.
 * $$a: V \setminus \{ v^0 \} \rightarrow \mathcal{A}$$ is an action partition corresponding each edge $$v$$ to a single action $$a(v)$$, fulfilling:

$$ \forall v \in \forall H \in \mathbf{H} $$, restriction $$a_v: s(v) \rightarrow A(H)$$ of $$a$$ on $$s(v)$$ is a bijection.
 * $$\mathcal{I} = \{ 1, ..., I \} $$ is a finite set of players, $$0$$ is (a special player called) nature, and $$[ \mathbf{H}_i)_{i \in I \cup \{ 0 \} }] $$ is a player partition of information set $$\mathbf{H}$$, (let $$\iota(v) = \iota(H)$$ be a single player that makes a move at node $$v \in H$$).
 * $$\rho = \{ \rho_H: A(H) \rightarrow [0, 1] | H \in \mathbf{H}_0 \}$$ is a family of probabilities of the actions of nature, and
 * $$u = (u_i)_{i \in \mathcal{I}}: T \rightarrow \Re^\mathcal{I}$$ is a payoff profile function.