Strategic interaction

A strategy’s value depends on the other player.

Compute best responses in a hawk-dove game, find a symmetric mixed equilibrium and separate equilibrium from collective welfare.

Start with: Expected value: multiply each payoff by its probability and add. A strategy is a rule for acting.

01 · Commit to a prediction

What do you expect?

Responses stay in this page only. Reloading or closing may discard them. Nothing is transmitted, saved or synchronized.

02 · Change an assumption

Predict. Change. Explain.

Before changing a control, say what should move and why. Start with the experiments below. Reset restores the starting model; it preserves your written responses.

Experiment 1

Keep V = 2 and C = 4. Change opponent Hawk share from 50% to 0%.

Check the prediction

Hawk gives 2, Dove gives 1. Hawk becomes the better response because fights never occur.

Experiment 2

Reset. Increase only C from 4 to 8.

Check the prediction

The equal-payoff opponent share falls from 1/2 to 1/4. A higher cost makes aggression attractive only when sufficiently rare.

03 · Connect the mechanism

From the picture to the quantities.

Hawk and Dove name strategies, not species or personalities. Two Hawks split the expected prize and expected fight cost equally. A Hawk facing a Dove takes the prize; two Doves split it. A best response maximizes one player’s expected payoff against a specified opponent distribution. At a symmetric interior mixed equilibrium, neither pure strategy has a payoff advantage.

U(H,H) = (V−C)/2; U(H,D) = V
U(D,H) = 0; U(D,D) = V/2
UH(p) − UD(p) = (V−pC)/2
Interior symmetric equilibrium p* = V/C when 0 < V < C

V and C are positive payoff quantities, 1 to 10 in the same arbitrary units. p is the chance an opponent plays Hawk, 0 to 1; controls show percent. U is expected payoff per encounter. If V ≥ C, all Hawk is a symmetric equilibrium. At V = C, a Dove against all Hawk ties rather than strictly loses.

Worked example

At V = 2, C = 4 and p = 1/2, Hawk earns (1/2)(−1) + (1/2)(2) = 0.5. Dove earns (1/2)(0) + (1/2)(1) = 0.5. Thus the 50/50 mixture is a symmetric Nash equilibrium. Two Doves would each earn 1, so equilibrium does not maximize their joint payoff.

Assumptions and limits

One-shot encounters, known payoffs, independent matching, no reputation and no repeated-game enforcement. The graph gives incentives, not a dynamic path to equilibrium. A mixed strategy can mean randomizing each encounter; a population mixture is another interpretation under suitable matching assumptions. No strategy is a moral recommendation, and this payoff matrix is not measured evidence about human conflict.

The annotated sources distinguish established results from this lesson’s original examples.

04 · Follow the structure

Where else does this apply?

Animal contests

Costs of fighting can make reproductive payoffs frequency dependent.

Boundary: Actual contests involve asymmetry, ownership, signals and history.

Shared-resource access

Aggressive access can win alone and collide when common.

Boundary: A communications protocol has different payoffs and enforceable rules; the biological analogy is limited.

05 · Retrieve without hints

Close the explanation. Try a new case.

Write an answer before opening its feedback. Later, return directly here without rereading above. Recognition, explanation and transfer are separate outcomes. No page action or answer reveal measures mastery.

Recognition and explanation

Reveal reasoning and rubric

Because Hawk’s fight cost occurs only against Hawk, while Dove’s shared prize occurs only against Dove. Different mixtures change expected returns.

Self-check: Name the conditional encounters and their probabilities.

Calculation

Reveal reasoning and rubric

p* = 1/2. Hawk: 0.5 × (−1.5) + 0.5 × 3 = 0.75. Dove: 0.5 × 1.5 = 0.75.

Self-check: Verify equality directly from the payoff matrix.

Novel transfer

Reveal reasoning and rubric

Not without a new model. Future rewards, memory and discounting change the available strategies and payoffs. The one-shot calculation remains valid only for one-shot assumptions.

Self-check: Identify the new strategic options and avoid treating equilibrium as a behavioral law.

On a later day, try again and record actual evidence in the curriculum. A later unaided explanation and a fresh transfer problem give stronger evidence than immediate familiarity. No reminder is scheduled.

Sources & scope

What supports the lesson?

Original teaching examples. Reference links need a connection; the lesson itself does not. Built 2026-10-11. Learner understanding is not assessed.