However, this solution is not stable, since if B believes A will choose A2 then B will choose B1 to gain 1 then if A believes B will choose B1 then A will choose A1 to gain 3 and then B will choose B2 and eventually both players will realize the difficulty of making a choice. Then, the maximin choice for A is A2 since the worst possible result is then having to pay 1, while the simple maximin choice for B is B2 since the worst possible result is then no payment. if the choices are A1 and B1 then B pays 3 to A). Assume the payoff matrix for B is the same matrix with the signs reversed (i.e. Suppose each player has three choices and consider the payoff matrix for A displayed on the table ('Payoff matrix for player A'). The following example of a zero-sum game, where A and B make simultaneous moves, illustrates maximin solutions. See also example of a game without a value.Įxample Payoff matrix for player A The name minimax arises because each player minimizes the maximum payoff possible for the other – since the game is zero-sum, they also minimize their own maximum loss (i.e. (a) Given Player 2's strategy, the best payoff possible for Player 1 is V, and (b) Given Player 1's strategy, the best payoff possible for Player 2 is − V.Įquivalently, Player 1's strategy guarantees them a payoff of V regardless of Player 2's strategy, and similarly Player 2 can guarantee themselves a payoff of − V. In the context of zero-sum games, the minimax theorem is equivalent to: įor every two-person, zero-sum game with finitely many strategies, there exists a value V and a mixed strategy for each player, such that In two-player zero-sum games, the minimax solution is the same as the Nash equilibrium. Its formal definition is: v i _ = max a i min a − i v i ( a i, a − i ) resulting from both players playing their maximin strategy. The maximin value is the highest value that the player can be sure to get without knowing the actions of the other players equivalently, it is the lowest value the other players can force the player to receive when they know the player's action. Originally formulated for several-player zero-sum game theory, covering both the cases where players take alternate moves and those where they make simultaneous moves, it has also been extended to more complex games and to general decision-making in the presence of uncertainty. When dealing with gains, it is referred to as 'maximin' – to maximize the minimum gain. Minmax (sometimes Minimax, MM or saddle point ) is a decision rule used in artificial intelligence, decision theory, game theory, statistics, and philosophy for minimizing the possible loss for a worst case ( maximum loss) scenario. For other uses, see Minimax (disambiguation). The other player moves all remaining seeds to their store, and the player with the most seeds in their store wins.This article is about the decision theory concept.

When one player no longer has any seeds in any of their houses, the game ends. There is no limit on the number of moves a player can make in their turn.Ħ. If the last sown seed lands in the player's store, the player gets an additional move. If the last sown seed lands in an empty house owned by the player, and the opposite house contains seeds, both the last seed and the opposite seeds are captured and placed into the player's store.ĥ. Moving counter-clockwise, the player drops one seed in each house in turn, including the player's own store but not their opponent's.Ĥ. On a turn, the player removes all seeds from one of the houses under their control. The player's score is the number of seeds in the store to their right.ģ. Each player controls the six houses and their seeds on the player's side of the board. At the beginning of the game, four (five to six) seeds are placed in each house.Ģ. The object of the game is to capture more seeds than one's opponent.ġ. The board has 6 small pits, called houses, on each side and a big pit, called an end zone or store, at each end. The game provides a Kalah board and a number of seeds or counters. It's an implementation of modern variant of mancala - kalah. There are a lot of games in the mancala family: oware, bao, omweso and so on. The objective is usually to capture all or some set of the opponent's pieces. The mancala games are a family of two-player turn-based strategy board games played with small stones, beans, or seeds and rows of holes or pits in the earth, a board or other playing surface.

Mancala game