How to mathematically express in a payoff matrix that "not losing" isn't the equivalent of winning

Question

My uncle was watching a documentary on the revolutionary war and one of the historians said, "Washington realized he didn't need to win the war, he only needed to not lose it." Is it possible to express in a pay off matrix that losing is not the antonym to winning?

Answer 1

The easiest way of doing this is saying that there are 2 levels of the game. Let $p_1$ and $p_2$ be two players, let $A$ be a finite set of all possible outcomes and let $f_1,f_2$ be a correspondent payoff matrix. Based on the outcome $A$, we determine new "meta" outcomes: $$ S_1 = \{f_1>f_2\},\; S_0 = \{f_1 = f_2\} = \{\text{draw}\}, \;S_2 = \{f_2>f_1\}. $$ Now, the final result of the game is evaluated not based on the first-level-payoffs $f_1,f_2$, but on the second-layer ones $u_1$ and $u_2$ which in your case are given by: $$ u_1(S_1) = u_1(S_0) = 1, \;u_1(S_2) = -1. $$ That means, the in the final count the player $p_1$ wants to avoid the loss in the first layer, but both the win and the draw in the first layer he considers as a final win.

P.S. this is highly informal, but seems to fit the question.