Information sets are primarily used in
extensive form representations of games and are typically depicted in
game trees. A game tree shows all possible paths from the start of a game to its various endings, with branches representing the choices available to players at each decision point. For games with imperfect information, the challenge lies in representing situations where a player cannot determine their exact position in the game. For example, in a card game, a player knows their own cards but not their opponent's cards, creating uncertainty about the true game state. This uncertainty is modeled using information sets. Information sets are typically represented in game trees using dotted lines connecting indistinguishable nodes, ovals encompassing multiple nodes, or similar notations indicating that a player cannot tell which of several positions they are actually in. This visual representation helps analyze how uncertainty affects optimal play.
Formal definition An information set in an extensive form game must satisfy the following properties: • Every node in the information set belongs to the same player. • The player cannot distinguish between any nodes within the same information set based on their available information. • All nodes in the same information set must have identical available actions. • No node in an information set can be an ancestor of another node in the same set (this would create a logical impossibility in the game timeline).
Strategic implications The structure of information sets profoundly affects strategic reasoning. When a player faces an information set with multiple nodes, they must formulate strategies that are optimal across all possible game states represented by that information set. This leads to several important game-theoretic concepts: •
Mixed strategies often become necessary when facing uncertainty, as pure strategies might be exploitable by opponents who can predict them. •
Bayesian updating occurs as players update their beliefs about which node in an information set they are at based on observed actions. •
Signaling and information revelation become strategic considerations, as players may take actions specifically to reveal or conceal information.
Dynamic games and backward induction In games with multiple information sets, the strategic interaction becomes dynamic rather than static. Players must reason not just about current decisions but about future information sets that might arise. The standard solution technique for such games is
backward induction, where players reason from the end of the game toward the beginning. For example, when player A chooses first, player B will make the best decision for themselves based on A's choice and their own information set at that time. Player A, anticipating this reaction, makes their initial choice to maximize their own payoff. This sequential reasoning process is complicated in games with imperfect information, requiring more sophisticated solution concepts like
sequential equilibrium that account for beliefs about which node in an information set a player is actually at. ==Example==