- One participant's gains are offset by losses from the other players.

- No value is created or destroyed.

- No value is created or destroyed.

Players Make decisions one after another:

- Tic-tac-toe

- Checkers

- Chess

These games can be represented by**Decision Tree**!

Solvable by backward induction, such as MinMax algorithm.

- Tic-tac-toe

- Checkers

- Chess

These games can be represented by

Solvable by backward induction, such as MinMax algorithm.

The choice of White to move is to MINIMIZE the MAXIMUM of potential reward by Black in the next move. In the picture above, the maximums of Black after each of possible White moves are (+3, +6, +2). Therefore, if White assumes Black is a sane player, White should minimize the possible gain by Black (+2) and choose to move knight.

Players Make decisions at the same time:

- Rock-paper-scissors

- Prisoner's Dilemma

- Final Jeopardy Wagering

These games can be represented in**Payoff Matrix**!

Solvable by probabilistic distribution & Nash equilibrium.

- Rock-paper-scissors

- Prisoner's Dilemma

- Final Jeopardy Wagering

These games can be represented in

Solvable by probabilistic distribution & Nash equilibrium.

- Two suspects in separate cells

- Neither cares about the other

- Each can either deny or confess:

* both deny: each gets 5 years

* both confess: each gets 10 years

* one of each: denier gets 20 years; confessor gets 0

Payoff Matrix:

- Neither cares about the other

- Each can either deny or confess:

* both deny: each gets 5 years

* both confess: each gets 10 years

* one of each: denier gets 20 years; confessor gets 0

Payoff Matrix:

Assume both A&B are sane agents:

- A think "confess" is better regardless of what B does, so A choose "confess". B knows that A would choose "confess", then B would also choose "confess".

- B think "confess" is better regardless of what A does, so B choose "confess". A knows that B would choose "confess", then B would also choose "confess".

This becomes a symmetric game, with both "confess" as its Nash Equilibrium:

- A think "confess" is better regardless of what B does, so A choose "confess". B knows that A would choose "confess", then B would also choose "confess".

- B think "confess" is better regardless of what A does, so B choose "confess". A knows that B would choose "confess", then B would also choose "confess".

This becomes a symmetric game, with both "confess" as its Nash Equilibrium:

We have a Nash Equilibrium!

- Leader is with 20000 points

- Trailer is with 14000 points

Suppose their answer to the Jeopardy question is completely random, meaning the likelihood of any of them get a question correct is 50%, so the likelihood of (Right, Right), (Right, Wrong), (Wrong, Right), (Wrong, Wrong) are equally likely of a 25% chance.

Seems there is a loop in the payoff matrix. Can we find a Nash equilibrium?

- A set of strategies such that no player can improve his situation by deviating.

- Not necessarily the best combined outcome for all players as a whole.

**Pure strategy**

- Do the same thing every time

- NOT guaranteed to have Nash Equilibrium)

**Mixed strategy**

- Choose action according to a probability distribution

- GUARANTEED to have Nash Equilibrium

- Calculation Steps (From trailer's perspective):

* Assign probability variable to each of the opponent's decisions.

* Determine payoffs for each of the opponent's decisions

* Find the value of the variable so that the payoffs are equivalent

]]>- Not necessarily the best combined outcome for all players as a whole.

- Do the same thing every time

- NOT guaranteed to have Nash Equilibrium)

- Choose action according to a probability distribution

- GUARANTEED to have Nash Equilibrium

- Calculation Steps (From trailer's perspective):

* Assign probability variable to each of the opponent's decisions.

* Determine payoffs for each of the opponent's decisions

* Find the value of the variable so that the payoffs are equivalent