Strategic games: separate best responses from a good outcome
A game model names who chooses, what each knows and how payoffs depend on everyone’s choices. This route separates equilibrium, coordination and a chosen measure of joint performance. A desirable outcome need not be selected automatically, and a device that correlates choices changes more than the players’ marginal frequencies.
Study route and exercise: University Reading List editorial project. How we curate · Sources checked:
On this page
Scope and responsibility
For learners comfortable with algebra and expected values. Games of Strategy is the introductory spine; Gibbons develops applied equilibrium reasoning; the free Algorithmic Game Theory collection is an advanced extension requiring algorithms and proof. Its free availability does not make it a beginner substitute. The exhibition game below is entirely invented.
Choose your starting point
I am learning strategic thinking
Use Games of Strategy to turn a verbal interaction into players, actions, information and a payoff table before selecting a solution concept.
I need to build applied economic models
Use Gibbons after intermediate microeconomics and probability. Check whether timing and information justify the equilibrium concept you use.
I need incentives plus computation or efficiency
Use selected chapters of the free Algorithmic Game Theory collection after the basics. Separate existence, finding an equilibrium and its welfare properties.
Compare the three books
Algorithmic Game Theory
Noam Nisan; Tim Roughgarden; Éva Tardos; Vijay V. Vazirani (editors) · 2007
Free advanced extension
- Choose it when
- You know equilibrium definitions and want computational, mechanism-design and efficiency questions.
- Use another route when
- You need your first account of a payoff table or lack algorithms/probability/proof preparation.
- Selected reading
- 2007 edited collection: introductory equilibrium material, then a named computation or efficiency chapter selected for your question.
- Limits and trade-offs
- Different contributors address different models and guarantees. The coeditor links an authorized complete reading route; free reading is not a blanket redistribution licence or a single linear introductory syllabus.
Try before choosing
For the exhibition table, distinguish the task of checking an equilibrium from finding all equilibria and comparing total payoffs.
Check your work: One efficient equilibrium does not show that every equilibrium is efficient or that players will select it.
Games of Strategy
Avinash K. Dixit; Susan Skeath; David McAdams · Edition: 6 · 2025
Introductory problem-solving spine
- Choose it when
- You want varied examples and accessible algebra before formal extensions.
- Use another route when
- You expect an everyday analogy alone to establish the assumptions of a real interaction.
- Selected reading
- Sixth edition,2025, ISBN 9781324087137: named simultaneous-choice, mixed-strategy and coordination discussions.
- Limits and trade-offs
- Dixit, Skeath and McAdams’s sixth edition differs from earlier author/edition combinations. The intuitive route still requires computing unilateral deviations rather than labeling an outcome Nash by inspection of the total.
Try before choosing
Mark every player’s best response in the original table before calculating any expected total.
Check your work: A best response compares one player’s own payoffs while holding the other’s action or distribution fixed.
Game Theory for Applied Economists
Robert Gibbons · 1992
Applied equilibrium-modeling spine
- Choose it when
- You want to translate economic situations into static/dynamic and complete/incomplete-information games.
- Use another route when
- Intermediate microeconomics, probability or the distinction between an action and a strategy is missing.
- Selected reading
- Gibbons 1992 paperback ISBN 9780691003955: static complete-information games first, then the timing/information class your application requires.
- Limits and trade-offs
- Its compact application-led argument is more demanding than an introductory strategy survey. The selected print and electronic identifiers differ; later-looking retail pages do not create a new edition.
Try before choosing
Explain why the simultaneous exhibition game and a move-observed sequential version are different models.
Check your work: Specify who observes what and when. An equilibrium concept does not transfer without checking that information structure.
Check your preparation
What is a Nash equilibrium?
A strategy profile where no player benefits from a unilateral change given the others’ strategies. It is not a promise of fairness, a maximum total or a prediction without model assumptions.
What is mixed here?
Each player independently randomizes before simultaneous action. Their joint probabilities are products; a common public random device can produce a different joint distribution.
Can payoffs be added?
Only because the packet stipulates comparable risk-neutral token payoffs. Their sum is an explicitly chosen classroom criterion, not a general measure of social welfare across people.
A six-step study plan
Steps completed: 0 / 6
These are your own study marks, not an assessment. They are not sent to this site.
-
Build the model
Selected reading: Games of Strategy simultaneous-choice introduction.
What to do: Write players, actions, timing, knowledge and the payoff order in each cell.
Check your work: A story condition cannot vanish when the table is drawn.
-
Check pure deviations
Selected reading: Introductory equilibrium reading.
What to do: Mark both players’ best responses to each pure action.
Check your work: Use each player’s own payoff, not the cell’s total.
-
Solve indifference
Selected reading: Selected mixed-strategy discussion.
What to do: Derive the two equations that make each player willing to mix.
Check your work: The opponent’s probability makes your actions indifferent; do not swap the roles of p and q.
-
Calculate joint outcomes
Selected reading: Probability review and the main packet.
What to do: List all four outcome probabilities, expected payoffs and mismatch probability.
Check your work: The probabilities sum to 1 and match the independent-product assumption.
-
Change the random device
Selected reading: The follow-up and selected coordination discussion.
What to do: Check obedience after each public signal and compare the joint distribution with equal marginals under independence.
Check your work: Correlation is not merely a different label for independent randomization.
-
State the limits
Selected reading: Gibbons modeling or the collection’s selected efficiency reading.
What to do: Write which equilibrium claim is proved and which selection/fairness/real-world claim is not.
Check your work: Keep verification, selection and a welfare criterion as separate conclusions.
Worked exercise
Original exhibition game: independent mixing can miss coordination
Two fictional exhibitors choose room X or Y simultaneously. Ordered token payoffs (A, B) are XX=(4,2), XY=(0,0), YX=(0,0), YY=(2,4). Find all pure and independently mixed Nash equilibria, then the mixed expected payoffs and mismatch probability.
Assumptions
Payoffs are common knowledge and comparable risk-neutral tokens. No transfers, communication, repetition or outside option occurs. Mixed choices are independent, with p=P(A chooses X) and q=P(B chooses X). This is a complete invented game, not advice for an actual negotiation.
- Against X each player strictly prefers X; against Y each strictly prefers Y. Thus XX and YY are pure equilibria, each total 6. Off-diagonal outcomes are not equilibria.
- A’s payoffs are 4 q from X and 2(1−q) from Y, so mixing needs q=1/3. B’s are 2 p and 4(1−p), so p=2/3. These give the sole interior independent mixed equilibrium; a pure opponent gives a strict pure best response, ruling out a one-player-only mixture.
- Joint probabilities XX=2/9, XY=4/9, YX=1/9, YY=2/9. Mismatch probability is 5/9; each player’s expected payoff is 4/3 and the total 8/3.
- Equilibrium means no profitable unilateral change, not maximal joint payoff or reliable coordination. The model does not select between its two pure equilibria.
Answer and interpretation
Pure equilibria XX and YY; independent mixed equilibrium p=2/3, q=1/3, expected 4/3 each, mismatch 5/9. The joint criterion is not maximized by that mixture.
Test the result
Check each pure deviation and both indifference equations, then sum the four joint probabilities. Independence is doing numerical work in this calculation.
Change an assumption
Changed institution: a common signal correlates the choices
Replace independent randomization with a public fair coin observed before actions: heads recommends both X, tails both Y. Keep the same payoffs and allow unilateral deviations after seeing the signal. Check obedience and expected payoffs. Compare independent play with the same 50/50 marginal frequencies.
- After heads, obeying X gives A 4 and B 2; a unilateral switch gives 0. After tails, obeying Y gives A 2 and B 4; switching again gives 0. Obedience is individually optimal after either signal.
- The joint distribution is XX = 1/2, YY = 1/2 with no off-diagonal mass. Each expected payoff is 3, total 6, and mismatch 0. This is a correlated device selecting pure equilibria.
- The same marginal probabilities under independence give each of four outcomes 1/4, expected 1.5 each and mismatch 1/2. That independent profile is not a Nash equilibrium: A prefers X and B prefers Y against a 50/50 opponent.
Answer and interpretation
The common signal supports obedience and expected 3 each. Equal marginals do not determine the joint distribution or its incentives.
Test the result
Verify the conditional deviation checks separately from ex ante averages. The coin’s availability and common observability are changed institutional assumptions, not conclusions inferred from the original table.
Common mistakes
Largest total called equilibrium
A profitable unilateral deviation is evaluated using one player’s payoff.
Own mixing probability used in own indifference
It is the opponent’s distribution that equalizes your pure-action payoffs.
Equal marginals imply independence
A shared random signal can put all probability on matching outcomes while retaining 50/50 individual frequencies.
Before moving on
- I can find and check every equilibrium claimed for the packet.
- I can reconstruct independent and correlated outcome tables.
- I can explain why equilibrium, selection and joint performance are different questions.
Sources and edition checks
Sources checked:
- Coeditor:2007 collection and authorized complete reading route
The 2007 collection extends established game-theory definitions toward computation and efficiency, with an authorized free route.
- Coauthor: Games of Strategy sixth edition 2025
The coauthor identifies the 2025 sixth edition and introductory strategy purpose.
- Duke publication record: exact selected sixth-edition ISBN
The institutional record fixes the selected ISBN and author combination.
- Princeton:1992 applied game theory book
The 1992 book develops model-building across timing and information classes. The exhibition and coin cases are original.
These are proposed study activities, not official course assignments or a claim that every book has been read cover to cover.