# Strategic games: separate best responses from a good outcome — Study worksheet [Study guides](https://unireadinglist.com/guides/strategic-games/) [Course topic: Game Theory](https://unireadinglist.com/subjects/economics/#course-game-theory) Sources checked: 2026-11-04 Study route and exercise: University Reading List editorial project. 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. Your notes: ________________________________ ISBN: 9780521872829 [Official free reading](https://www.cs.cmu.edu/~sandholm/cs15-892F13/algorithmic-game-theory.pdf) [Book record and access options](https://unireadinglist.com/subjects/economics/algorithmic-game-theory-nisan/) ### Games of Strategy Avinash K. Dixit; Susan Skeath; David McAdams · 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. Your notes: ________________________________ ISBN: 9781324087137 [Book record and access options](https://unireadinglist.com/subjects/economics/games-of-strategy/) ### 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. Your notes: ________________________________ ISBN: 978-0-691-00395-5 [Book record and access options](https://unireadinglist.com/subjects/economics/game-theory-for-applied-economists/) ## Check your preparation ### What is a Nash equilibrium? Your notes: ________________________________ ### What is mixed here? Your notes: ________________________________ ### Can payoffs be added? Your notes: ________________________________ ## A six-step study plan - [ ] 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. Your notes: ________________________________ - [ ] 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. Your notes: ________________________________ - [ ] 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. Your notes: ________________________________ - [ ] 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. Your notes: ________________________________ - [ ] 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. Your notes: ________________________________ - [ ] 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. Your notes: ________________________________ ## 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. Your notes: ________________________________ ## 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. Your notes: ________________________________ ## 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. ## Answer and interpretation ### 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. ### Original exhibition game: independent mixing can miss coordination 1. 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. 2. 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. 3. 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. 4. Equilibrium means no profitable unilateral change, not maximal joint payoff or reliable coordination. The model does not select between its two pure equilibria. 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 1. 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. 2. 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. 3. 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. 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. ## Sources and edition checks - [Coeditor:2007 collection and authorized complete reading route](https://timroughgarden.org/books.html) The 2007 collection extends established game-theory definitions toward computation and efficiency, with an authorized free route. - [Coauthor: Games of Strategy sixth edition 2025](https://sites.duke.edu/mcadams/) The coauthor identifies the 2025 sixth edition and introductory strategy purpose. - [Duke publication record: exact selected sixth-edition ISBN](https://scholars.duke.edu/publication/1676754) The institutional record fixes the selected ISBN and author combination. - [Princeton:1992 applied game theory book](https://press.princeton.edu/books/paperback/9780691003955/game-theory-for-applied-economists) 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. [How we curate](https://unireadinglist.com/editorial/)