Game of survival

A two-person dynamic game with final pay-off taking values 0 and 1 only. Thus, the terminal set $X^T$ is split into two subsets $X^{T+}$ and $X^{T-}$, where player I wins if the game hits a state $x \in X^{T+}$, while II wins if it hits a state $X \in X^{T-}$. If the game is non-terminating, player I wins, and player II loses, some number $h_\infty \in [0,1]$. If $h_\infty = 0$, one has a game of survival for player II, while if $h_\infty = 1$ one has a game of survival for player I.

Historically, the concept of a game of survival goes back to the classical "gamblers ruin" problem. A direct generalization of this problem is the game in which at every stage one and the same play matrix is performed, while changes in state are expressed as changes in the fortunes of the participants (see [1]). Player I wins if his opponent is ruined (that is, if the opponent's fortune becomes negative), and loses if he is ruined himself. Such a game has a value not depending on $h_\infty$, and both players have stationary $\epsilon$-optimal strategies if all entries in the matrix of the iterated subgame are non-zero (cf. also Strategy (in game theory)). In this case the game is almost certainly completed in a finite number of steps. Another variant of games of survival (with multi-component fortunes) are the so-called "games to exhaustion" (see [2]). Differential games of survival can be looked at as a generalization of the games under discussion to the case of continuous time.

References

[1]	J.W. Milnor, L.S. Shapley, "On games of survival" , Contributions to the theory of games , 3 , Princeton Univ. Press (1957) pp. 15–45
[2]	D. Blackwell, "On multi-component attrition games" Naval. Res. Logist. Quart. , 1 (1954) pp. 210–216
[3]	I.V. Romanovskii, "Game-type random walks" Theor. Probabl. Appl. , 6 (1961) pp. 393–396 Teor. Veroyatnost. i Primenen. , 6 : 4 (1961) pp. 426–429

References

[a1]	R.D. Luce, H. Raiffa, "Games and decisions. Introduction and critical survey" , Wiley (1957)