> For the complete documentation index, see [llms.txt](https://theshank.gitbook.io/ai/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://theshank.gitbook.io/ai/game-theory-and-ml/lecture-2-22-01.md).

# Lecture 2 - 22/01

{% embed url="<https://docs.google.com/presentation/d/17xPJWGJeAWkIWI8x7L8zl7mIek681lGpn5MCnydFB2A/edit#slide=id.gab813e4041_0_0>" %}

## Prisoner's Dilemma

If agents do not cooperate, the best (global) outcome possible is missed.

## COP 21 Game

* N governments
* 2 actions per states
  * Do not pollute (Cost = 3)
  * Pollute (cost=1 and +1 for everyone)

What is the equilibrium? &#x20;

## Multi-Player Game

* Siimultaneous move games
* n players, each player pick a **strat** and occurs a loss.

$$
l\_k(s\_1, ...s\_n) = l\_k(s\_k, s\_{-k})
$$

**Goal of the player: Minimize their loss**

## Zero-Sum Two-player Games

Zero-sum: $$\sum\_{k=1}^n l\_k = 0$$

n=2

Action for each players: $$i \in \[n] = {1,....,n}$$ and $$j \in \[m]$$

### Game

$$
\min\_{i\in\[n]} \max\_{j \in \[m]} l\_{ij}
$$

### Mix strategies

For example in the game of rock-paper-scissor

We have probabilities over actions of each player as $$p=\[p1, p2, .... p\_n] \in \Delta\_n$$and $$q=\[q1,q2, ..., q\_m]  \in \Delta\_m$$

$$\Delta\_n := {p \in R^n: p\_1+...p\_n=1, p\_i >= 0}$$

Payoff: $$l(p,q):= E\_{i\sim p, j \sim q} \[l\_{ij}] = p^TLq$$

**Game:** $$\min\_{p\in \Delta\_n} \max\_{q \in \Delta\_m} p^TLq$$

## **Nash Equilibrium of a Game**

**Best worst-case move**

$$
s^\* \in \text{NASH}  \implies l\_k(s^*\_k, s^**{-k}) \leq l\_k(s\_k, s^\**{-k}) \forall s
$$

#### **Theorem**&#x20;

Any game with a finite set of players and a finite set of strategies has a Nash equilibrium of mixed strategies.&#x20;

\`