All materials come from Coursera《Game Theory》by Stanford University & The University of British Columbia

## Intro to nash equilibrium

Attributes of nash equilibrium

• nobody has an incentive to deviate** from their actions if a nash equilibrium is achieved**
• each player's action maximizes his/her payoff given the actions of others
• a self-consistent or stable profile

## Pure Strategy Nash Equilibrium

### Pure Strategy And Mixed Strategy

the informal definitions of pure strategy and mixed strategy are

• pure strategy: only one action is played with positive probability
• mixed strategy: more than one action is played with positive probability

With a pure strategy, a player choose one action with 100% ensurance. While with a mixed strategy, a player can have a probability of 50% choosing action A and a probability of 50% choosing action B. In another word, player $i$ chooses action $a_i$ randomly from $A$ with corresponding probability.

### Best Response

The idea of Best Response: if you know what everyone else was going to do, it would be easy to pick your own action.

Based on the previous lecture, $a$ represents action profile, which organized by $n$ players' action

$a = (a_1, a_2,..., a_n)$

Now we denote $a_{-i}$, which means a sub action profile of other players act

$a_{-i} = (a_1, a_2,..., a_{i-1}, a_{i+1},..., a_n)$

Then we can re-define $a$ as

$a = (a_i, a_{-i})$

Therefore, the definition of Best Response is showen below

$a_i^* \in BR(a_{-i}) \ \ iff \ \ \forall a_i \in A_i, u_i(a_i^*, a_{-i}) \ge u_i(a_i, a_{-i})$

That is to say, given $a_{-i}$, $a_i^*$ is the best response for player $i$ to play.

But in practice, no agent knows what the others will do, so we can't say about what actions will actually occur. The idea here is to look for stable action profiles, leading to equilibrium point. $a = (a_1, a_2,..., a_n)$ is a (pure strategy) nash equilibrium iff $\forall i,\ a_i \in BR(a_{-i})$

### Dominant Strategy

generalize from actions to strategies