Value and optimal strategies of a zero-sum matrix game

A pure saddle point is detected and returned exactly. Otherwise fictitious play runs for a FIXED number of rounds and a rigorous BRACKET is returned with the estimate: the value lies between min_j (x'A)_j and max_i (Ay)_i.

Usage

Matgame(A, iters = 2000)

Arguments

A

Payoff matrix to the ROW player.

iters

Fixed number of fictitious-play rounds.

Value

List with value, lower, upper, row_strategy, col_strategy, maximin, minimax, saddle, iterations, m, n.

Details

Formula: v = max_x min_y x' A y = min_y max_x x' A y

References

von Neumann, J. (1928), Zur Theorie der Gesellschaftsspiele, Mathematische Annalen 100, 295-320 – the minimax theorem. The fictitious-play iteration is Brown (1951) and its convergence for zero-sum games is Robinson (1951), Annals of Mathematics 54(2), 296-301; neither is von Neumann's and both are cited to their own sources.

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
Matgame(V)
#> $value
#> [1] 8
#> 
#> $lower
#> [1] 8
#> 
#> $upper
#> [1] 8
#> 
#> $row_strategy
#> [1] 0 0 0 0 0 0 0 1
#> 
#> $col_strategy
#> [1] 1
#> 
#> $maximin
#> [1] 8
#> 
#> $minimax
#> [1] 8
#> 
#> $saddle
#> [1] 1
#> 
#> $iterations
#> [1] 0
#> 
#> $m
#> [1] 8
#> 
#> $n
#> [1] 1
#> 
#> $method
#> [1] "Matrix game with a pure saddle point"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"