This week’s game theory was dedicated to introduction, overview, uses of game theory, some applications and examples, and formal definitions of: the normal form, payoffs, strategies, pure strategy Nash equilibrium, dominant strategies..

Define a Game

  1. Normal form: List what payoffs get as a function of their actions.
  2. Extensive form: Includes timing of moves, players moves sequentially, represented as a tree.

Finite, n-person normal form game: $<n, a,=”” u=””>$:</n,>

  • Players: $N={1, \ldots, n}$, indexed by i;
  • Action set for player i: $a=(a_1, \ldots, a_n) \in A = A_1 \times \ldots \times A_n$ is an action profile;
  • Utility function or Payoff function for player i: $u_i:A \mapsto \mathbf{R} $, $u=(u_1, \ldots, u_n)$ is a profile of utility functions.

Type of Games

Type of Game Properties Examples
Pure Competition 1. Exactly two players of opposed interests; Zero sum special case when $u_1(a)+u_2(a)=0$ Matching Pennies, Rock-Paper-Scissors
Coordination Players have same interests: $\forall a \in A, \forall i,j, u_i(a)=u_j(a)$ side of road
Coordination and Competition Battle of the Sexes

Nash Equilibrium


Someone has an incentive to deviate from a profile of actions that do not form an equilibrium.

Best Resopnse
: If you knew what everyone else was going to do, it would be easy to pick your own action.
: Nash equilibrium looks for stable action profiles.

Dominant Strategies

Strategy (currently) is choosing an action (“pure strategy”)

Denote $s_i$ and $s_i’$ as two strategies for player i, and $S_{-i}$ be the set of all possible strategy profiles for the other players.

$s_i$ strictly dominates $s_i’$ if $ \forall s_{-i} \in S_{-i}, u_{i}(s_i, s_{-i}) \gt u_{i}(s_i’, s_{-i})$
$s_i$ very weakly dominates $s_i’$ if $ \forall s_{-i} \in S_{-i}, u_{i}(s_i, s_{-i} ) \ge u_{i}(s_i’, s_{-i})$
Please pay attention to the difference between best response, which lies in the definition of strategy.

A strategy profile consisting of dominant strategies for every player must be a Nash equilibrium! An equilibrium in strictly dominant strategies must be unique.

Pareto Optimality

Some times, one outcome $o$ is at least as good for every agent as another outcome $o’$, and there’s some agent who strictly prefers $o$ to $o’$.

An outcome $o^*$ is Pareto-optimal if there is no other outcome that Pareto-dominates it.

[^1]: Nash equilibrium,