Extensive-form game


In game theory, an extensive-form game is a specification of a game allowing for the explicit representation of a number of key aspects, like the sequencing of players' possible moves, their choices at every decision point, the 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.

Finite extensive-form games

Some authors, particularly in introductory textbooks, initially define the extensive-form game as being just a game tree with payoffs, and add the other elements in subsequent chapters as refinements. Whereas the rest of this article follows this gentle approach with motivating examples, we present upfront the finite extensive-form games as constructed here. This general definition was introduced by Harold W. Kuhn in 1953, who extended an earlier definition of von Neumann from 1928. Following the presentation from, an n-player extensive-form game thus consists of the following:
  • A finite set of n players
  • A rooted tree, called the game tree
  • Each terminal node of the game tree has an n-tuple of payoffs, meaning there is one payoff for each player at the end of every possible play
  • A partition of the non-terminal nodes of the game tree in n+1 subsets, one for each player, and with a special subset for a fictitious player called Chance. Each player's subset of nodes is referred to as the "nodes of the player".
  • Each node of the Chance player has a probability distribution over its outgoing edges.
  • Each set of nodes of a rational player is further partitioned in information sets, which make certain choices indistinguishable for the player when making a move, in the sense that:
  • * there is a one-to-one correspondence between outgoing edges of any two nodes of the same information set—thus the set of all outgoing edges of an information set is partitioned in equivalence classes, each class representing a possible choice for a player's move at some point—, and
  • * every path in the tree from the root to a terminal node can cross each information set at most once
  • the complete description of the game specified by the above parameters is common knowledge among the players
A play is thus a path through the tree from the root to a terminal node. At any given non-terminal node belonging to Chance, an outgoing branch is chosen according to the probability distribution. At any rational player's node, the player must choose one of the equivalence classes for the edges, which determines precisely one outgoing edge except the player doesn't know which one is being followed. A pure strategy for a player thus consists of a selection—choosing precisely one class of outgoing edges for every information set. In a game of perfect information, the information sets are singletons. It's less evident how payoffs should be interpreted in games with Chance nodes. It is assumed that each player has a von Neumann–Morgenstern utility function defined for every game outcome; this assumption entails that every rational player will evaluate an a priori random outcome by its expected utility.
The above presentation, while precisely defining the mathematical structure over which the game is played, elides however the more technical discussion of formalizing statements about how the game is played like "a player cannot distinguish between nodes in the same information set when making a decision". These can be made precise using epistemic modal logic; see for details.
A perfect information two-player game over a game tree can be represented as an extensive form game with outcomes. Examples of such games include tic-tac-toe, chess, and infinite chess. A game over an expectminimax tree, like that of backgammon, has no imperfect information but has moves of chance. For example, poker has both moves of chance and imperfect information.

Perfect and complete information

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. The labels by every edge of the graph are the name of the action that edge represents.
The initial node belongs to player 1, indicating that player 1 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:,, and. The payoffs associated with each outcome respectively are as follows,, and.
If player 1 plays D, player 2 will play U' to maximise their payoff and so player 1 will only receive 1. However, if player 1 plays U, player 2 maximises their 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.

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. An information set is a set of decision nodes such that:
  1. Every node in the set belongs to one player.
  2. When the game reaches the information set, the player who is about to 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 right is the same as the above game except that player 2 does not know what player 1 does when they come to play. The first game described has perfect information; the game on the right 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, play in the first game will be as follows: player 1 knows that if they play U, player 2 will play D' and so player 1 will receive 2. However, if player 1 plays D, player 2 will play U' and player 1 will receive 1. Hence, in the first game, the equilibrium will be 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 they have played U when they have 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 1 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.
To more easily solve this game for the Nash equilibrium, it can be converted to the normal form. Given this is a simultaneous/sequential game, player one and player two each have two strategies.
  • Player 1's Strategies:
  • Player 2's Strategies:
Up' Down'
Up
Down

We will have a two by two matrix with a unique payoff for each combination of moves. Using the normal form game, it is now possible to solve the game and identify dominant strategies for both players.
  • If player 1 plays Up, player 2 prefers to play Down
  • If player 1 plays Down, player 2 prefers to play Up
  • If player 2 plays Up, player 1 prefers to play Down
  • If player 2 plays Down, player 1 prefers to play Down
These preferences can be marked within the matrix, and any box where both players have a preference provides a nash equilibrium. This particular game has a single solution of with a payoff of.
In games with infinite action spaces and imperfect information, non-singleton information sets are represented, if necessary, by inserting a dotted line connecting the endpoints behind the arc described above or by dashing the arc itself. In the Stackelberg competition 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.