Dominant Strategies

1. What is a Dominant Strategy?

Dominant Strategy

A strategy that gives the best payoff regardless of what the opponent does.

Strategy $s_i$ strictly dominates $s'_i$ if:

$$u_i(s_i, s_{-i}) > u_i(s'_i, s_{-i}) \quad \forall s_{-i}$$

Strictly Dominant

Always gives strictly higher payoff than any other strategy, no matter what opponents play.

$$u_i(s_i, s_{-i}) \; \color{#32D583}{>} \; u_i(s'_i, s_{-i})$$

for ALL $ s_{-i} $

Weakly Dominant

Gives at least as high payoff (≥), with strictly higher for at least one opponent strategy.

$$u_i(s_i, s_{-i}) \; \color{#d39e00}{\geq} \; u_i(s'_i, s_{-i})$$

for ALL $ s_{-i} $

Notation: $ u_i $ = utility (payoff) function for player $ i $ | $ s_i $ = strategy of player $ i $ | $ s_{-i} $ = strategies of all other players

Key Insight

If you have a dominant strategy, just play it! You don't need to think about what the opponent will do — your best choice is the same regardless.

2. Interactive: Shop Price War Analysis

How to Find a Dominant Strategy (Methodology)

1

Fix opponent's strategy to option A

2

Compare your payoffs for each of YOUR strategies

3

Repeat for opponent's option B

4

If same strategy wins BOTH times → Dominant!

Finding Shop A's Dominant Strategy

1

2

3

4

Shop A's Profit | Shop B's Profit

		Shop B
		High $	Low $
Shop A	High $	3, 3	-2, 5
Shop A	Low $	5, -2	-1, -1

(3, 3): Both high → steady profits

(-1, -1): Both low → price war!

(5, -2): A undercuts → A wins big

(-2, 5): B undercuts → B wins big

Step 1: When Shop B sets HIGH price, what should Shop A do?

When Shop B sets High $

A: High $

3

vs

A: Low $

5

When Shop B sets Low $

A: High $

-2

vs

A: Low $

-1

Analysis Summary

When B sets High $ → A should set: ?

When B sets Low $ → A should set: ?

Complete both comparisons to find dominant strategy...

3. Dominated Strategies & IESDS

Dominated Strategy

A strategy that is always worse than some other strategy.

$s_i$ is strictly dominated by $s'_i$ if: $u_i(s'_i, s_{-i}) > u_i(s_i, s_{-i}) \quad \forall s_{-i}$

Rational players never play dominated strategies!

IESDS: Iterated Elimination of Strictly Dominated Strategies

1

Identify any strictly dominated strategy

2

Eliminate it (rational players won't play it)

3

In reduced game, look for newly dominated

4

Repeat until no more elimination possible

IESDS Interactive Example: 3×3 Game

The Goal

Systematically eliminate strategies that no rational player would ever choose. Each elimination may reveal new dominated strategies!

P1's Payoff | P2's Payoff

		Player 2
		L	M	R
P1	U	4, 3	5, 1	6, 2
	M	2, 1	8, 4	3, 6
	D	3, 0	9, 6	2, 8

Crossed out = Eliminated (dominated)

Click "Eliminate Next" to start the analysis

Elimination Steps

Step 1: ?

Step 2: ?

Step 3: ?

Current Analysis

Look at each strategy and compare: Is there another strategy that ALWAYS gives a better payoff?

4. Dominant Strategy Equilibrium

Dominant Strategy Equilibrium (DSE)

An outcome where every player is playing a dominant strategy.

Very strong solution concept

Doesn't require beliefs about opponents

Rare in practice - most games lack DSE

Prisoner's Dilemma Paradox

(Confess, Confess) is the DSE, giving -3, -3

But (Silent, Silent) gives -1, -1 — better for both!

Individual rationality → collective suboptimality

When DSE is Good

DSE is easy to predict and compute
No coordination problem — each player acts independently
Used in mechanism design (e.g., second-price auctions)

Auction design aims to create dominant strategies!