Technical Treatment

Definition

Given a non-cooperative game of $x$ players $ \{p_{1},p_{2},...p_{x}\} $ with a defined range of action $A_{i}$ available for players; and a utility function $f$ that computes result of the game when $a_{i}\in A_{i}$, The stag hunt described above can be described as the following:

  • A set of players $P$, here $P = \{p_{1}, p_{2}\}$
  • $A_{i}: \{S, R\}$ where $S$ stands for stag and $R$, rabbit
  • $f = \sum u_{i}:$ $u$ is a player's utility score.
  • Action profile can thus be described as such: $\{SS, RR, SR\}$.
  • For $\{SS\}$, $f = 4$ each hunter get 2 units. For $\{SR\}$, $f = 1$ stag hunter gets a utility of 1 units while the other gets none. Finally, for $\{SS\}$,they both will receive an utility of 1, i.e. $f = 2$

Nash Equilibria

This concept will allow us to define an optimal strategy for this game. It can be explained as such. If $p_{1}$ performs $a_{1}$ and $p_{1}$ doesn't change action profile after been informed of $a_{2}$(and vice versa), we say that we reach a state of Nash equilibria.

More precisely I will say:

$$f(a_{1}, a_{2}) \geq (a_{1}^{'}, a_{2}) $$$$f(a_{1}, a_{2}) \geq (a_{1}, a_{2}^{'})$$

Here the equilibrium is obvious.