Topic 3 — Game Theory

Part of: Microeconomics Strategic Interaction, Normal-Form Games, Dominated Strategies, Nash Equilibrium, Mixed Strategies Micro 3 — Tiomkin School of Economics, Reichman University Key concepts: Game Theory · Nash Equilibrium · Dominated Strategy · Normal-Form Game · Mixed Strategy · Best Response · Prisoner's Dilemma · Coordination Game

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1. What is Game Theory?

Game Theory provides formal tools to model and analyse strategic situations — situations where:

• Each player's outcome depends not just on their own actions, but on the actions of others.
• Rational players must account for others' behaviour when deciding what to do.

Definition: A Game

A game is a formal description of a strategic situation. It specifies the players, their available strategies, and the payoffs they receive for every possible strategy combination.

Classic example: Two firms in the same industry — each firm's profit depends on both its own decisions (hiring, pricing, advertising) and its rival's decisions. Neither can choose optimally without considering what the other will do.

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2. Normal-Form (Matrix) Representation

The normal form is the standard way to represent a simultaneous game — all strategies and payoffs in a payoff matrix.

Reading a Payoff Matrix

• Rows = strategies for Player 1
• Columns = strategies for Player 2
• Each cell = (payoff to P1, payoff to P2)
• The first number is always Player 1's payoff

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3. Types of Simultaneous Games

3.1 Zero-Sum (Competitive) Game

One player's gain is exactly the other's loss. Payoffs in every cell sum to zero.

Example — Odd/Even:

Intuition

In zero-sum games there is pure conflict — what's good for one player is bad for the other. There is no scope for cooperation.

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3.2 Battle of the Sexes

A coordination game with conflicting preferences — both players prefer to coordinate, but disagree on where.

Example: A couple chooses between Football (F) and Ballet (B). Player 1 prefers Football, Player 2 prefers Ballet — but both prefer being together over being apart.

Key feature

There are two Nash Equilibria in pure strategies: (F, F) and (B, B). The players face a coordination problem — which equilibrium will they land on?

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3.3 Coordination Game

Both players want to coordinate, and one outcome is Pareto-dominant (better for both), but there's also a "safe" but worse equilibrium.

Version 1 — Simple:

Version 2 — With catastrophic downside:

Common exam point

In Version 2, (A, A) is Pareto-dominant but (B, B) is risk-dominant — the "safe" choice, because deviating to A when the other plays B is catastrophic (payoff = −100). Exam questions often ask you to identify both equilibria and comment on which is more likely to emerge.

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3.4 Marketing Competition / Prisoner's Dilemma

Two firms decide whether to hire a marketing agent or not.

This is the structure of the Prisoner's Dilemma

• Mutual cooperation (Not hire, Not hire) → (5, 5): the collectively best outcome.
• But each firm has an incentive to deviate: "Hire" always yields more regardless of the opponent's choice (6 > 5; 4 > 3).
• The Nash Equilibrium is (Hire, Hire) → (4, 4), which is worse for both than (Not hire, Not hire) → (5, 5).
• "Hire" is a dominant strategy for each player.

Intuition — The Prisoner's Dilemma Tragedy

Individual rationality leads to a collectively irrational outcome. Neither player can trust the other to cooperate, so both defect and end up worse off. This explains why cartels need enforcement, countries need treaties, etc.

Comparison: Game Types

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4. Dominated Strategies

Definition: Strictly Dominated Strategy

Strategy $s_i$ is strictly dominated by $s_i'$ if, for every possible strategy of the opponent, $s_i'$ gives a strictly higher payoff: $u_i(s_i', s_{-i}) > u_i(s_i, s_{-i}) \quad \forall \, s_{-i}$ A rational player will never play a strictly dominated strategy.

Definition: Weakly Dominated Strategy

Strategy $s_i$ is weakly dominated by $s_i'$ if $s_i'$ does at least as well in all cases and strictly better in at least one: $u_i(s_i', s_{-i}) \geq u_i(s_i, s_{-i}) \quad \forall \, s_{-i}, \quad \text{with strict inequality for some } s_{-i}$

Common mistake

Don't confuse dominant strategy with best response. A dominant strategy beats all others regardless of what the opponent does. A best response is only optimal given a specific opponent strategy. Every dominant strategy is a best response, but not vice versa.

Worked Example — Marketing Competition

For Player 1:

• If P2 plays "Not hire": Hire gives 6 > Not hire gives 5 ✓
• If P2 plays "Hire": Hire gives 4 > Not hire gives 3 ✓

→ "Hire" strictly dominates "Not hire" for Player 1. By symmetry, the same holds for Player 2.

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5. Iterated Elimination of Strictly Dominated Strategies (IESDS)

Since rational players never play dominated strategies — and they know their opponents are rational too — we can iteratively eliminate dominated strategies until none remain.

IESDS Algorithm

1. Find any strictly dominated strategy for any player and eliminate it.
2. Repeat on the reduced game.
3. Stop when no strictly dominated strategies remain. The order of elimination does not affect the final result.

Key Theorem

A Nash Equilibrium is never eliminated by IESDS. If IESDS yields a unique outcome, that outcome is the unique Nash Equilibrium. Use this as a sanity check.

Worked Example — 4×4 Game

How to approach IESDS

1. For the row player (P1): For each pair of rows, check if one gives a strictly higher payoff across all columns. If yes, eliminate the dominated row.
2. For the column player (P2): For each pair of columns, check if one gives a strictly higher payoff across all rows. If yes, eliminate the dominated column.
3. Repeat on the smaller matrix until nothing more can be eliminated.

Exam shortcut

When checking whether row $X$ dominates row $Y$ for Player 1: go column by column and check if $X > Y$ in every column. If even one column has $Y \geq X$, it's not strict dominance.

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6. Nash Equilibrium

Definition: Nash Equilibrium

A Nash Equilibrium (NE) is a strategy profile $(s_1^*, s_2^*, \ldots, s_n^*)$ such that no player can improve their payoff by unilaterally deviating, given the strategies of all others: $\boxed{u_i(s_i^*, s_{-i}^*) \geq u_i(s_i, s_{-i}^*) \quad \forall \, s_i, \; \forall \, i}$

Intuition

At a Nash Equilibrium, every player is playing a best response to what the others are doing. There is no regret: knowing everyone else's strategy, no one wants to change theirs unilaterally.

How to Find Nash Equilibria — Best Response Method

1. For each column, find the row that gives Player 1 the highest payoff → underline that payoff.
2. For each row, find the column that gives Player 2 the highest payoff → underline that payoff.
3. Any cell where both payoffs are underlined is a Nash Equilibrium.

Best Response Method — Prisoner's Dilemma

• P1's BR: "Hire" is best regardless of P2 (6 > 5; 4 > 3) → underline 6 and 4 in "Hire" row.
• P2's BR: "Hire" is best regardless of P1 (6 > 5; 4 > 3) → underline 6 and 4 in "Hire" column.
• Nash Equilibrium: (Hire, Hire) → (4, 4) ✓

Nash Equilibria for Each Game

Prisoner's Dilemma:

Only one NE, even though (Not hire, Not hire) → (5, 5) is Pareto-superior. This is the dilemma.

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Battle of the Sexes:

Two pure-strategy NE. Coordination problem: each player prefers a different one.

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Coordination Game (with catastrophe):

(A, A) is Pareto-dominant; (B, B) is risk-dominant.

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7. Mixed Strategy Nash Equilibrium

Not every game has a pure-strategy Nash Equilibrium. When players randomise over their strategies, we get a mixed-strategy equilibrium.

Definition: Mixed Strategy

A mixed strategy assigns a probability distribution over pure strategies. Player $i$ plays strategy $s_i^k$ with probability $p_k$, where $\sum_k p_k = 1$.

Key Condition for a Mixed NE

At a mixed-strategy NE, a player must be indifferent between all strategies played with positive probability — otherwise they would deviate entirely to the strictly better one: $EU_i(s_i^k) = EU_i(s_i^j) \quad \text{for all strategies played with positive probability}$

Classic Example — Rock, Paper, Scissors

Is there a pure-strategy NE?

No — R → P → S → R → ... Best responses cycle and never settle. No pure NE exists.

type: payoff-matrix
rows: Rock,Paper,Scissors
cols: Rock,Paper,Scissors
payoffs: 0,0;-1,1;1,-1|1,-1;0,0;-1,1|-1,1;1,-1;0,0

Why the cycle rules out a pure equilibrium

Whatever your opponent commits to, your best response beats it — but then their best response beats yours, and so on around the loop. No strategy profile is stable, so there can be no pure-strategy Nash equilibrium. The only equilibrium is to randomise uniformly, $(\tfrac13,\tfrac13,\tfrac13)$, which leaves your opponent indifferent and unable to exploit you.

Mixed-strategy NE:

Verification — if P2 plays $(\frac{1}{3}, \frac{1}{3}, \frac{1}{3})$, Player 1's expected payoffs are:

Player 1 is indifferent → any mixing is a best response → $(\frac{1}{3}, \frac{1}{3}, \frac{1}{3})$ is rational. ✓

Mixed NE in Battle of the Sexes (beyond pure NE)

Let Player 1 play F with probability $p$, Player 2 play F with probability $q$.

Make P1 indifferent (P2 chooses $q$): $EU_1(F) = EU_1(B) \implies 2q = 1(1-q) \implies 3q = 1 \implies q = \tfrac{1}{3}$

Make P2 indifferent (P1 chooses $p$): $EU_2(F) = EU_2(B) \implies 1 \cdot p = 2(1-p) \implies 3p = 2 \implies p = \tfrac{2}{3}$

Mixed NE: P1 plays F with prob $\frac{2}{3}$; P2 plays F with prob $\frac{1}{3}$.

type: mixed-strategy-br

All three equilibria in one picture

Each axis is a player's probability of choosing Football. The blue and red step-functions are the two players' best responses: a player jumps from "definitely Ballet" to "definitely Football" once the other is likely enough to play Football. The correspondences intersect in three places — the two corners are the pure equilibria (F,F) and (B,B), and the interior crossing at $(p=\tfrac23,\,q=\tfrac13)$ is the mixed equilibrium. The mixed NE is the one where each player makes the other exactly indifferent.

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8. Solution Concepts — Summary Table

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9. Practice Questions & Worked Answers

Q1 — Find all Nash Equilibria

Game:

Method — underline best responses:

• P1 vs col L: max(3, 2) = 3 → underline U. P1 vs col R: max(1, 4) = 4 → underline D.
• P2 vs row U: max(2, 4) = 4 → underline R. P2 vs row D: max(1, 3) = 3 → underline R.

Nash Equilibrium: (D, R) → (4, 3). Only cell with both underlined. ✓

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Q2 — Identify dominated strategies

Is "U" dominated in the game above?

• vs L: U gives 3, D gives 2 → U > D ✓
• vs R: U gives 1, D gives 4 → D > U ✗

→ U is not dominated (it wins in one column). Neither strategy is strictly dominated here.

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Q3 — Apply IESDS step by step

Game:

Step 1 — Player 2's columns: Does C dominate L? P2 gets: C=(3,1,2) vs L=(3,4,2). Row M: 1 < 4 ✗. No. Does R dominate C? P2 gets: R=(2,3,4) vs C=(1,3,2). All R ≥ C, and Row T: 2>1, Row B: 4>2 → R weakly dominates C. Eliminate C (if using weak dominance — check if exam allows).

Step 1 (strict only) — Player 1's rows: Does T dominate B? T=(4,2,3) vs B=(2,1,4). Col R: 3 < 4 ✗. No. Does M dominate B? M=(3,3,2) vs B=(2,1,4). Col R: 2 < 4 ✗. No.

Note

Always state whether you're using strict or weak dominance — the exam may specify. Strict dominance is the default.

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Q4 — Solve for the Mixed-Strategy NE

Find the mixed NE of:

Step 1: Find $q$ (prob P2 plays L) to make P1 indifferent:

Set equal: $3 - 3q = 2q \implies 3 = 5q \implies q = \tfrac{3}{5}$

Step 2: Find $p$ (prob P1 plays U) to make P2 indifferent:

Set equal: $2p = 3 - 3p \implies 5p = 3 \implies p = \tfrac{3}{5}$

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Q5 — Essay: Why is the Prisoner's Dilemma NE Pareto-Inefficient, and How Can Players Escape It?

Model answer:

In the Prisoner's Dilemma, "Hire" is a dominant strategy for both players — it yields a higher payoff regardless of the opponent's action (6 > 5 if the other cooperates; 4 > 3 if the other defects). Rational players therefore both choose "Hire", yielding payoffs (4, 4).

This outcome is Pareto-inefficient because an alternative allocation — (Not hire, Not hire) → (5, 5) — makes both players strictly better off. The problem is that individual rationality and collective rationality diverge: following the dominant strategy is individually optimal but collectively disastrous.

Escaping the Prisoner's Dilemma requires changing the structure:

• Repeated interaction (Folk Theorem): If the game is played infinitely (or with uncertain end), cooperation can be sustained via strategies like "Tit for Tat" — cooperate first, then mirror the opponent's last move. The threat of future punishment disciplines defection.
• Binding contracts: If players can make enforceable pre-game commitments (e.g., via a regulator or legal agreement), they can credibly commit to cooperating.
• Internalised social preferences: If payoffs include the opponent's welfare (altruism or guilt from defection), the dominant strategy may shift toward cooperation.
• Communication: Pre-play "cheap talk" can coordinate expectations in some settings, though it is not binding.

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10. Summary

Game theory provides the tools to analyse any situation where outcomes are interdependent. From this topic:

• Normal form: represent games as payoff matrices; rows = P1 strategies, columns = P2 strategies, cells = (P1 payoff, P2 payoff).
• Dominated strategies: never played by rational agents. Eliminate them iteratively (IESDS) to narrow the strategy space.
• Nash Equilibrium: a strategy profile where no one wants to deviate. Find it by underlining best responses — doubly underlined cells are NE.
• Multiple NE: games can have 0, 1, or many pure-strategy NE (but always at least one mixed-strategy NE in finite games).
• Prisoner's Dilemma: dominant strategies lead to a Pareto-inferior outcome — escaping requires repeated play, contracts, or preference changes.
• Mixed-strategy NE: solve by making the opponent indifferent between their strategies.

Exam Checklist

1. Write out the payoff matrix clearly.
2. Underline best responses — find pure NE (doubly underlined cells).
3. Check for dominant strategies — if one exists, it's always played.
4. Apply IESDS if asked — be systematic, check strict dominance column by column.
5. For mixed NE: set $EU_i(s^k) = EU_i(s^j)$ and solve for the opponent's mixing probability.
6. Sanity check: NE must survive IESDS — if it doesn't, re-check your work.

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Related Notes

• Microeconomics — subject hub
• Topic 1 - Asymmetric Information
• Topic 2 - Equilibrium in Different Market Structures