Game Theory

1/20/2026

Board

Ann has two option
- Stop
  - Bob has options
    - 1
      - Ann gets another turn
        She can choose up
        Game ends payouts are (2,7,4,1)
        She can choose down
        (1,-2,3,0)
    - 2
      - Deb has a turn
        High
        1,3,2,-11,3
        Low
        0,-2,-7,-8,
    - 3
      - 10,7,1,1
- Go
  - Leads to Chris’s options:
    - Risky
      - Nature - a player but does not get a payoff.
        Good outcome
        50% probability (6,3,4,0)
        Bad outcome
        50% probability (2,8,-1,2)
    - Safe
      - (3,5,3,1)
Payoffs:

Numerical
Generally higher is better
Interpersonal Comparison
Expected payoff
1. Nature has this move
  1. Good
    1. Ann will get a 6
  2. Bad
    1. Ann will get a 2
2. (0.5)(6) + (0.5)(2)=4

Move & Strategy
- Move: Single action
- Strategy: Plans for the succession of moves that players expect to make in all of the various eventualities that might arise in the course of the game.

Solving Games by using trees

Board

Carmen
- Try smoking
  - Continue to smoke
    - Payoff is -1
  - Don’t smoke
    - Payoff is 1
- Don’t try smoking
  - no payoff
Carmen (Today)
- Try Smoking
  - Future Carmen (new player)
    - Continue smoking
      - (-1, 1) = payoff
        Past Carmen has -1 payoff
    - Stop smoking
      - (1, -1)
        Past Carmen has 1 payoff
- Don’t smoke
  - no payoff

Steps

Find Equilibrium
(Example) The decision to smoke

Rollback Equilibrium

Method of looking ahead and reasoning back to determine behavior in a sequential moved game
1. Also known as backward induction
The set of strategies when all players choose their optimal strategies found by rollback analysis.
1. Smoking game
  1. Rollback equilibrium would be: Today’s Carmen chooses the strategy of no or not.
    1. The optimal strategy for future Carmen is to continue smoking.

Emily

Contribute
- Nina
  - (Contribute) Talia
    - Contribute (3,3,3)
    - Don’t Contribute (3,3,4)
  - (Don’t Contribute) Talia
    - Contribute (3,4,3)
    - Don’t Contribute (1,2,2)
Don’t Contribute
- Nina
  - (Contribute) Talia
    - Contribute 4,3,3
    - Don’t Contribute 2,1,2
  - (Don’t Contribute) Talia
    - Contribute
      - Talia
        Contribute (2,2,1)
        Don’t Contribute (2,2,2)

3 Things:

Available strategies for each player
- Emily: 2 strategies
  - Contribute
  - Don’t Contribute
- Nina: 4 strategies
  - Contribute, Don’t Contribute
  - Contribute, Contribute
  - Don’t Contribute, Contribute
  - Don’t Contribute, Don’t Contribute
- Talia: 15 strategies
  - Contribute, contribute, contribute, contribute
  - contribute, contribute, contribute, don’t contribute
  - contribute, contribute, don’t contribute, contribute
Optimal Strategy
Actual path of play

1/22/2026

3 Stages Problem

What do we want?
Who makes it?
Who gets it?

Example

Entrepreneur
- Invest
  - Next move goes to nature
    - Succeed(p)
      - Player 2 (new player)
        Buy from entrepreneur
    - Don’t succeed (1-p)
      - Outcome -1
- Don’t invest
  - Outcome: 0

2/3/2026

Centipede Game

Player A
- Pass
  - Player B
    - Pass
      - Player A
        Pass
        Take
        Ends (30,0)
    - Take
      - Ends (0,20)
- Take
  - Ends (10,0)

2/5/2026

Game Table (Also called Normal Form)

Discrete Strategies

Nash Equilibrium (NE)

Definition: A list of strategies, one for each player, such that no player can get a better payoff by switching to some other strategy that is available to them while all other players adhere to the strategies specified for them in the list.

Nash Equilibrium Strategies

Cell by cell
1. Looking at literally every space
Dominance (Prisoners Dilemma)
1. This is technically classified as a non-cooperative game.
2. Best Response Analysis
  1. Even for both sides
Successive Elimination of dominated strategies
1. The really long one knocking row by row and column by column.
Best Response Analysis
1. Circling number by number fast
  1. Examining each choice, either takes the longest or shortest amount of time.

2/10/2026

Coordination Games

Pun Coordination

Search Strategies
- Dominance
- Cell by cell
- BRA

2/24/2026

2 stage games and sub games

Example: Crosstalkk and global dialogue

invest: $10billion

if neither invest = end of game

if 1 invest and other doesn’t = pricing decision

high price: 60 million + 3.14 = $400

low price: 80 million + 3.14 = $200

if both invest = 2nd simultaneous

2,-2 6,-6

30,-30 -10, 10

3/10/2026

Pure Strategies and Continuous Variables

Price Competition
1. Setting:
  1. X’s: Tapas Bar
    1. Each has a set menu
    2. Each has to set a price for menu
    3. Goal: maximize π (=paying in game per week)
    4. Each makes their choice simultaneously
  2. Cost: $8 to serve customer
  3. Variables:
    1. Px = X’s Price
    2. Py = Y’s price
    3. Qx = # of X’s customer
    4. Qy = # of Y’s customer
  4. Equations:
    1. Qx = 44-2Px + Py
    2. Qy = 44-Py + Px

3/26/2026

Mixed Strategy:

Definition: Random mixtune between given pure strategies

A: Tennis Game

Continuous game of mixed strategies

4/2/2026

Threats
Promises
Commitments
- 3 above this lead to 2-stage games
  - Leads to 2st stage
    - 2nd stage = original game
Observable, irreversible
Side-payment
- Dan (Player) - Node A
  - Pass - Leads to node B
    - Patrick
      - Pass
        11,11
      - Take
      - 10,12
  - Take
    - 1,-10

How can Dan get Patrick to commit to choosing “Pass” at node B

-Take-it-or-leave-it offer: “If you promise to choose “Pass at Node B, I’ll give you a payment of 2, otherwise I’ll play “Pass”.