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