2 Summary
3 Dominant Strategies & The Prisoner’s Dilemma
A dominant strategy is one that is best for a player regardless of what the other players does.The most famous example is the Prisoner’s Dilemma.
3.1 The Classic Prisoner’s Dilemma
Two prisoners are separated and interrogated. Each must choose:- Confess (Rat)- Deny (Stay Silent) If both stay silent, they receive light sentences. If one confesses and the other stays silent, the confessor is rewarded and the silent prisoner receives a harsh sentence. If both confess, they receive moderate sentences. The dominant strategy for each is to confess, even though both would prefer the outcome where they remain silent. This logic also applies to:- Joint projects where working is costly - Environmental agreements - Price competition - Public‑goods situations. A dominated strategy is one that always yields a lower payoff than another. In the Prisoner’s Dilemma, staying silent is dominated. The tragedy is that individual incentives destroy collective welfare.
4 Nash Equilibrium
A Nash equilibrium is a set of strategies where no player wants to deviate unilaterally.
At equilibrium, each player’s choice is a best response to the other’s.
4.1 Payoff Matrices
A typical 2×2 game is represented as:
| Player 2: L | Player 2: R | |
|---|---|---|
| Player 1: U | (a, b) | (c, d) |
| Player 1: D | (e, f) | (g, h) |
We search for strategy pairs where each player is doing the best possible given the other’s choice.
4.2 Examples
4.2.1 Hawk–Dove (a conflict game)
Both players benefit if one yields, but not if both fight.
Multiple Nash equilibria may exist.
4.2.2 Games from the notes:
- Price competition
- Basic battle outcomes
- Economic policy decisions (interest rates, spending)
Nash equilibrium captures strategic stability:
no one has an incentive to move away from the current outcome. ``
5 Coordination & Anti‑Coordination Games
Some games require alignment rather than conflict.
5.1 Coordination Game
Two students must choose the same “flat tire” explanation when questioned. Success depends on coordinating answers.
Coordination problems arise when:
- Driving conventions (left vs right)
- Technology adoption
- Business standards
- Social norms
Failure to coordinate leads to poor outcomes.
5.2 Battle of the Sexes
Two players want to meet but prefer different activities.
There are two Nash equilibria, reflecting mutual desire to coordinate.
Focal points help players select between multiple equilibria, relying on:
- Tradition
- Logic
- Cultural expectations
5.3 Matching Pennies (Anti‑Coordination)
A pure conflict game where the players want opposite outcomes.
There is no pure‑strategy Nash equilibrium.
This motivates mixed strategies, covered in the next chapter. ``
6 Mixed Strategies
When no pure‑strategy Nash equilibrium exists, players may randomize.
A mixed strategy assigns probabilities to each pure action.
6.1 Example: Penalty Kick
The kicker chooses left or right.
The keeper dives left or right.
If the keeper can perfectly predict the kicker’s choice, the kicker always loses. Thus the kicker must randomize.
Let:
- p = probability the keeper dives left
- (1 − p) = probability the keeper dives right
The kicker chooses probabilities that make the keeper indifferent. The keeper does the same.
The equilibrium occurs when both choose each side with probability 1/2.
6.2 Battle of the Sexes (Mixed)
When multiple pure equilibria exist, there is also a mixed equilibrium determined by equating expected payoffs.
Mixed strategies reflect:
- unpredictability
- strategic concealment
- balancing incentives
They are fundamental in competitive environments such as sports, auctions, and military conflict.
7 Sequential Games & Backward Induction
Sequential games involve moves made over time, with later players observing earlier actions.
The key tool is backward induction:
Start from the final decision and reason backward.
7.1 Game Trees
A game tree includes:
- decision nodes
- branches representing actions
- terminal nodes with payoffs
Examples:
7.1.1 1. One‑on‑One Poker
A player bets, the opponent folds or calls.
Backward induction identifies credible strategies.
7.1.2 2. Investment Project
Player 1 invests funds; Player 2 either uses them properly or steals.
If the temptation is too high, stealing becomes rational, preventing investment.
Solutions include:
- collateral
- profit‑sharing
- small project size
These modify incentives to restore cooperation.
Sequential games also help model commitment:
By burning bridges (literally in one example), a player removes future options to make current threats credible. ``
8 Repeated Games & Cooperation Over Time
Many strategic interactions repeat over time.
Repeated games allow cooperation even when one‑shot incentives favor defection.
8.1 Trigger Strategies
Two important strategies:
- Tit for Tat — start cooperative, then copy the opponent’s last move
- Grim Trigger — cooperate until the opponent defects once, then punish forever
8.2 Finite Horizon
If the game is repeated a known number of periods, backward induction collapses the entire game: defection occurs every round.
8.3 Infinite or Indefinite Horizon
If players are uncertain when the game will end, cooperation becomes sustainable.
Let:
- δ = probability the game continues another round (discount factor)
Cooperation is sustainable when: Cooperative payoff ≥ Deviation payoff + Future punishment loss
This yields inequalities like:
δ ≥ threshold
The exact threshold depends on payoffs.
Repeated games explain:
- reputation
- trust
- collusion
- environmental agreements
- long‑run cooperation
9 Applications & Classroom Games
This chapter explores real‑world and classroom applications of game theory.
9.1 1. The Pizza Contribution Game
Three students decide whether to contribute money for pizza.
If at least two contribute, all benefit.
If few contribute, the public good fails.
This creates:
- multiple Nash equilibria
- free‑riding incentives
- first‑mover advantages
9.2 2. Pollution & Abatement
Countries choose whether to pollute or abate.
Pollution is individually tempting but collectively harmful.
This is a Prisoner’s Dilemma variant.
9.3 3. Price Competition
Firms can choose high or low prices.
Low prices may dominate, leading to competitive races.
9.4 4. The Guess‑2/3‑of‑the‑Average Game
Students choose numbers from 1 to 100.
Winner is closest to 2/3 of the class average.
Iterated reasoning leads to the Nash equilibrium of 1.
9.5 5. The Lions and Sheep Puzzle
A chain of lions consider eating the sheep.
Backward induction determines which lions act.
9.6 6. Bridge‑Burning in Invasion Games
A country commits to fighting by destroying its retreat path.
Removing the option to retreat changes equilibrium behavior.
Game theory highlights how changing the rules changes the outcome. ``