strictly-competitive-games-and-secuirity-strategies.mdx

title	date	tags	draft	summary
Strictly Competitive Games and Security Strategies	2022-7-12	game thoery competitive-games security-strategies	false	Introduction to Competitive Games and Security Strategies

Strictly competitive game

• A two-player game is strictly competitive if, for any two strategy profiles $s$ and $s'$, $u_1(s) \geq u_1(s')$ if and only if $u_2(s) \leq u_2(s')$.
• $s_i$ is a security strategy for player $i$ if it solves:

$$\max_{s_i \in S_i} \min_{s_j \in S_j} u_i(s)$$

• Note that a secure strategy might not be rationalizable.
• Player i's security level is the corresponding payoff.

Result

• If $s^$ is a nash equilibrium of a strictly competitive game, then $s^_1$ and $s^*_2$ are security strategies for player 1 and player 2.

Example

	F	C	B
F	0, 5	2, 3	2, 3
C	2, 3	0, 5	3, 2
B	5, 0	3, 2	2, 3

• We first find the security strategies for player 1 and player 2. For player 1, playing B would be the security strategy and for player 2, playing C or B will be the secure strategy. (Why?) Thus, the secure startegy set is, ${(B, C), (B, B)}$

'''mermaid graph TD; A-->B; A-->C; B-->D; C-->D; '''