two player zero-sum game

Definition

A two-player game is a zero-sum game if for each pair of strategies $(s_I, s_{II})$ one has

u_I (s_I, s_{II}) + u_{II} (s_I, s_{II}) = 0

Notes

Assuming that the players have von Neumann-Morgenstern utilities, any player's utility function is only determined only up to a positive affine transformation.

Maxmin and minmax strategies

As payoffs satisfy $u_I + u_{II} = 0$ , we may focus on one function $u_1 = u$ , $u_{II} = -u$ . Suppose player 1 (P1) seeks to maximize, and player 2 (P2) seeks to minimize (note: this convention is reversed in some textbooks).

Then, P1's maxmin value is given by

v_I = \max_{s_I \in S_I} \min_{s_{II} \in S_{II}} u(s_I, s_{II})

and P2's maxmin value is

v_{II} = \min_{s_{II} \in S_{II}} \max_{s_I \in S_I} u(s_I, s_{II})

Security level and security strategy

See theorem, security levels of matrix game players

\min_i a_{ij^*} = \underline{V}(A) \leq \overline{V}(A) = \max_j a_{i^* j}

Finite two-person zero-sum game

Game where

player set: $N = \{1,2\}$
action set $A_i$ for player $i$ : $a = (a_1, a_2) \in A = A_1 \times A_2$ , $A_1$ and $A_2$ are finite sets
utility function (or payoff function) for player ii:
- $u_i: A \mapsto \mathbb{R}$
- $u = (u_1, u_2)$ such that $u_1(a) + u_2(a) = 0$ for all $a \in A$

Notes

It is a zero-sum game and a game of pure competition

References

M. Maschler, E. Solan, and Shmuel Zamir, Game Theory, Cambridge University Press, 2013, pp. 111-116.
T. Başar and G.J. Olsder, Dynamic Noncooperative Game Theory, 2nd edition, Classics in Applied Mathematics, SIAM, Philadelphia, 1999.
https://bpb-us-e1.wpmucdn.com/wp.nyu.edu/dist/5/2123/files/2019/12/Lecture-2-Scribe.pdf