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Constant sum: A game is a constant sum game if the sum of the payoffs to every player are the same for every single set of strategies. In these games, one player gains if and only if another player loses. A constant sum game can be converted into a zero sum game by subtracting a fixed value from all payoffs, leaving their relative order unchanged.
The result is that both players are guaranteed to win – which is absurd, thus contradicting the assumption that such a strategy exists. Strategy-stealing was invented by John Nash in the 1940s to show that the game of hex is always a first-player win, as ties are not possible in this game. [2]
Separately, game theory has played a role in online algorithms; in particular, the k-server problem, which has in the past been referred to as games with moving costs and request-answer games. [125] Yao's principle is a game-theoretic technique for proving lower bounds on the computational complexity of randomized algorithms , especially online ...
S is α-effective if the members of S have strategies s.t. no matter what the complement of S does, the outcome will be a. S is β-effective if for any strategies of the complement of S, the members of S can answer with strategies that ensure outcome a. Finite game is a game with finitely many players, each of which has a finite set of strategies.
In game theory, a strategy A dominates another strategy B if A will always produces a better result than B, regardless of how any other player plays no matter how that player's opponent or opponents play. Some very simple games (called straightforward games) can be solved using dominance.
A classic example of a dynamic game with types is a war game where the player is unsure whether their opponent is a risk-taking "hawk" type or a pacifistic "dove" type. Perfect Bayesian Equilibria are a refinement of Bayesian Nash equilibrium (BNE), which is a solution concept with Bayesian probability for non-turn-based games.
A game with perfect information may or may not have complete information. Poker is a game of imperfect information, as players do not know the private cards of their opponents. Games where some aspect of play is hidden from opponents – such as the cards in poker and bridge – are examples of games with imperfect information. [5] [6]
In the mathematical area of game theory and of convex optimization, a minimax theorem is a theorem that claims that (,) = (,)under certain conditions on the sets and and on the function . [1]