Common errors — #equilibria

Frequent mistakes when setting up, solving, or reading game-theoretic models; quick diagnostic prompts and the right framing instead.

FIELD GUIDE

Common errors

Recurring traps from setting up and analysing strategic interactions. Each item is the wrong framing, the diagnostic question, and the correction.

Setting up the game

Conflating “action” with “strategy”

Wrong: “Player 1’s strategies are cooperate and defect.” Diagnostic: Is there more than one decision node, or any private information? Right: A strategy is a complete contingent plan over every information set. In a one-shot game with a single decision, strategies = actions. In a repeated or extensive-form game, strategies are functions of history/type.

Treating a sequential game as simultaneous

Wrong: Writing a normal-form payoff matrix for what is actually an extensive-form game with observable moves. Diagnostic: Does the second player observe the first player’s choice before acting? Right: Use the extensive form and apply backward induction. Normal-form analysis discards the timing information and often produces equilibria that aren’t subgame-perfect.

Forgetting beliefs in incomplete information

Wrong: Solving a signalling game as if both players have full information and computing pure Nash equilibria. Diagnostic: Is any payoff-relevant variable known to one player but not the other? Right: Specify types, prior, and belief updating. Use Perfect Bayesian Equilibrium; require beliefs consistent with Bayes’ rule wherever the equilibrium reaches the information set.

Computing equilibria

Asserting a unique Nash equilibrium

Wrong: “The Nash equilibrium of this 2×2 game is (Cooperate, Cooperate).” Diagnostic: Did you check for mixed-strategy equilibria and coordination failures (multiple pure NE)? Right: Enumerate all pure NE first, then test for mixed-strategy NE via indifference conditions. Coordination games typically have two pure NE plus one mixed NE.

Mixing strategies that aren’t best-responses

Wrong: A “mixed equilibrium” in which a player randomises across actions that have unequal expected payoffs. Diagnostic: Does the mixed strategy’s support contain only actions yielding equal expected payoff against the opponent’s strategy? Right: A player only randomises in equilibrium across actions that are indifferent. Pure best-responses outside the support must yield strictly lower payoff.

Confusing subgame perfection with Nash

Wrong: Treating any Nash equilibrium of an extensive-form game as the solution. Diagnostic: Does the equilibrium prescribe a strategy that involves a non-credible threat in some unreached subgame? Right: Restrict to subgame-perfect equilibria — those that remain Nash in every subgame.

Off-equilibrium beliefs left undefined

Wrong: Writing a PBE that doesn’t specify what the receiver believes at off-path information sets. Diagnostic: Does your equilibrium reach every information set with positive probability? Right: If an information set is off-path, beliefs there are unrestricted by Bayes’ rule — you must specify them and they must be such that the strategy is sequentially rational. Different off-path beliefs can sustain different equilibria; this is why refinements (intuitive criterion, D1, divinity) exist.

Repeated and dynamic games

Discount-factor handwaving

Wrong: “Cooperation is sustainable if players are patient enough.” Diagnostic: What is the exact threshold $\delta^*$, derived from the deviation payoff comparison? Right: Write the equality between cooperation’s discounted stream and the one-shot deviation gain plus punishment. Solve for $\delta^*$. State the threshold explicitly.

Trigger strategy without credible punishment

Wrong: “Defection triggers permanent defection by everyone” — without checking that the punishment phase is itself a Nash equilibrium. Diagnostic: Is the punishment strategy profile a Nash equilibrium of the stage game? Right: Verify the punishment is incentive-compatible. Grim trigger works because Nash reversion is itself an SPE; more elaborate punishments (Abreu-Pearce-Stacchetti) need their own verification.

Folk-theorem misuse

Wrong: “By the folk theorem, any payoff is achievable in equilibrium.” Diagnostic: Is the proposed payoff individually rational (above each player’s minmax) and feasible (a convex combination of stage-game payoffs)? Right: The folk theorem says individually rational and feasible payoffs are sustainable, not arbitrary ones. Compute minmax values and the feasibility set before invoking.

Auctions and mechanism design

Revenue equivalence outside its assumptions

Wrong: Citing revenue equivalence between formats with risk-averse bidders, asymmetric value distributions, or correlated values. Diagnostic: Are bidders risk-neutral, values independently drawn from the same distribution, and the lowest-type bidder’s expected payoff zero? Right: Revenue equivalence is a theorem of symmetric IPV auctions with risk neutrality. Deviating along any axis breaks it: under risk aversion, first-price gives higher revenue than second-price.

Truthful bidding in first-price auctions

Wrong: “In a first-price auction with private values, bidders bid their value.” Diagnostic: Does bidding $v_i$ leave any rent on the table for the winner? Right: Optimal bid-shading: $b_i^*(v_i) = \mathbb{E}[V_{(n-1)} \mid V_{(n-1)} < v_i]$ in symmetric IPV. Bidders bid the expected second-highest value, conditional on winning.

Confusing VCG with second-price

Wrong: Treating VCG as “second-price auction generalised to multi-unit settings.” Diagnostic: Does the mechanism charge each winner the externality they impose on others? Right: VCG charges winner $i$ the difference between the welfare others would have obtained without $i$ and the welfare others do obtain at the optimal allocation including $i$. In single-item settings this collapses to the second-highest bid; in multi-unit settings it generally does not.

Cooperative game theory

Shapley value vs core membership

Wrong: “The Shapley value is in the core.” Diagnostic: Is the game convex ($v(S \cup T) + v(S \cap T) \ge v(S) + v(T)$)? Right: Shapley value lies in the core for convex games. In non-convex games it may fall outside; the core may even be empty while the Shapley value still exists.

Empty core misdiagnosed as “no solution”

Wrong: “The core is empty, so the game has no solution.” Diagnostic: Have you considered the nucleolus, kernel, bargaining set, or Shapley value? Right: Many solution concepts coexist; an empty core simply rules out stable coalitional payoff vectors with the strict-domination property. The nucleolus always exists for games with non-empty imputation set.

Statistics and inference inside game-theoretic studies

Treating QRE as a noise model

Wrong: Adding logit noise to Nash predictions to fit experimental data without re-solving for fixed points. Diagnostic: Are your QRE predictions a fixed point of the logit best-response operator, not a perturbation of Nash? Right: QRE is solved as a fixed point of the noisy best-response map. The fixed point can be qualitatively different from any “perturbed Nash” because all players’ noise interacts.

Power calculations after the fact

Wrong: “We didn’t reject; let’s compute post-hoc power to argue the test was insufficient.” Diagnostic: Was the power calculation done before data collection, at the smallest effect size of substantive interest? Right: Power must be planned ex ante. Post-hoc “observed power” is a deterministic transformation of the p-value and adds no inferential content.

Reading and reporting

Equilibrium ≠ prediction

Wrong: Reporting a model’s equilibrium as the prediction for real-world behaviour without acknowledging alternatives. Diagnostic: Did the equilibrium concept assume best-response or rationality your subjects might not satisfy? Right: Distinguish equilibrium as benchmark from equilibrium as point prediction. Behavioural game theory (QRE, level-$k$, learning) supplies disciplined alternatives.

Strategy ≠ outcome

Wrong: “The Nash equilibrium of this game is the payoff vector $(2, 2)$.” Diagnostic: Did you describe a strategy profile, or just the payoffs at one? Right: A Nash equilibrium is a strategy profile $(s_1^*, \dots, s_n^*)$. The payoff vector $(u_1(s^*), \dots, u_n(s^*))$ is the outcome at that equilibrium, not the equilibrium itself.

--- title: "Common errors — #equilibria" description: "Frequent mistakes when setting up, solving, or reading game-theoretic models; quick diagnostic prompts and the right framing instead." toc: true toc-depth: 2 page-layout: full --- ```{=html} <div class="hero"> <div class="kicker">FIELD GUIDE</div> <h1>Common errors</h1> <p class="lead">Recurring traps from setting up and analysing strategic interactions. Each item is the wrong framing, the diagnostic question, and the correction.</p> </div> ``` ## Setting up the game ### Conflating "action" with "strategy" **Wrong**: "Player 1's strategies are *cooperate* and *defect*." **Diagnostic**: Is there more than one decision node, or any private information? **Right**: A strategy is a complete contingent plan over every information set. In a one-shot game with a single decision, strategies = actions. In a repeated or extensive-form game, strategies are functions of history/type. ### Treating a sequential game as simultaneous **Wrong**: Writing a normal-form payoff matrix for what is actually an extensive-form game with observable moves. **Diagnostic**: Does the second player observe the first player's choice before acting? **Right**: Use the extensive form and apply backward induction. Normal-form analysis discards the timing information and often produces equilibria that aren't subgame-perfect. ### Forgetting beliefs in incomplete information **Wrong**: Solving a signalling game as if both players have full information and computing pure Nash equilibria. **Diagnostic**: Is any payoff-relevant variable known to one player but not the other? **Right**: Specify types, prior, and belief updating. Use Perfect Bayesian Equilibrium; require beliefs consistent with Bayes' rule wherever the equilibrium reaches the information set. ## Computing equilibria ### Asserting a unique Nash equilibrium **Wrong**: "The Nash equilibrium of this 2×2 game is (Cooperate, Cooperate)." **Diagnostic**: Did you check for *mixed-strategy* equilibria and *coordination* failures (multiple pure NE)? **Right**: Enumerate all pure NE first, then test for mixed-strategy NE via indifference conditions. Coordination games typically have two pure NE plus one mixed NE. ### Mixing strategies that aren't best-responses **Wrong**: A "mixed equilibrium" in which a player randomises across actions that have unequal expected payoffs. **Diagnostic**: Does the mixed strategy's support contain only actions yielding equal expected payoff against the opponent's strategy? **Right**: A player only randomises in equilibrium across actions that are indifferent. Pure best-responses outside the support must yield *strictly lower* payoff. ### Confusing subgame perfection with Nash **Wrong**: Treating any Nash equilibrium of an extensive-form game as the solution. **Diagnostic**: Does the equilibrium prescribe a strategy that involves a non-credible threat in some unreached subgame? **Right**: Restrict to subgame-perfect equilibria — those that remain Nash in every subgame. ### Off-equilibrium beliefs left undefined **Wrong**: Writing a PBE that doesn't specify what the receiver believes at off-path information sets. **Diagnostic**: Does your equilibrium reach every information set with positive probability? **Right**: If an information set is off-path, beliefs there are unrestricted by Bayes' rule — *you must specify them* and they must be such that the strategy is sequentially rational. Different off-path beliefs can sustain different equilibria; this is why refinements (intuitive criterion, D1, divinity) exist. ## Repeated and dynamic games ### Discount-factor handwaving **Wrong**: "Cooperation is sustainable if players are patient enough." **Diagnostic**: What is the exact threshold $\delta^*$, derived from the deviation payoff comparison? **Right**: Write the equality between cooperation's discounted stream and the one-shot deviation gain plus punishment. Solve for $\delta^*$. State the threshold explicitly. ### Trigger strategy without credible punishment **Wrong**: "Defection triggers permanent defection by everyone" — without checking that the punishment phase is itself a Nash equilibrium. **Diagnostic**: Is the punishment strategy profile a Nash equilibrium of the stage game? **Right**: Verify the punishment is incentive-compatible. Grim trigger works because Nash reversion is itself an SPE; more elaborate punishments (Abreu-Pearce-Stacchetti) need their own verification. ### Folk-theorem misuse **Wrong**: "By the folk theorem, any payoff is achievable in equilibrium." **Diagnostic**: Is the proposed payoff individually rational (above each player's minmax) and feasible (a convex combination of stage-game payoffs)? **Right**: The folk theorem says *individually rational and feasible* payoffs are sustainable, not arbitrary ones. Compute minmax values and the feasibility set before invoking. ## Auctions and mechanism design ### Revenue equivalence outside its assumptions **Wrong**: Citing revenue equivalence between formats with risk-averse bidders, asymmetric value distributions, or correlated values. **Diagnostic**: Are bidders risk-neutral, values independently drawn from the same distribution, and the lowest-type bidder's expected payoff zero? **Right**: Revenue equivalence is a theorem of *symmetric* IPV auctions with risk neutrality. Deviating along any axis breaks it: under risk aversion, first-price gives higher revenue than second-price. ### Truthful bidding in first-price auctions **Wrong**: "In a first-price auction with private values, bidders bid their value." **Diagnostic**: Does bidding $v_i$ leave any rent on the table for the winner? **Right**: Optimal bid-shading: $b_i^*(v_i) = \mathbb{E}[V_{(n-1)} \mid V_{(n-1)} < v_i]$ in symmetric IPV. Bidders bid the expected second-highest value, conditional on winning. ### Confusing VCG with second-price **Wrong**: Treating VCG as "second-price auction generalised to multi-unit settings." **Diagnostic**: Does the mechanism charge each winner the externality they impose on others? **Right**: VCG charges winner $i$ the difference between the welfare others *would* have obtained without $i$ and the welfare others *do* obtain at the optimal allocation including $i$. In single-item settings this collapses to the second-highest bid; in multi-unit settings it generally does not. ## Cooperative game theory ### Shapley value vs core membership **Wrong**: "The Shapley value is in the core." **Diagnostic**: Is the game *convex* ($v(S \cup T) + v(S \cap T) \ge v(S) + v(T)$)? **Right**: Shapley value lies in the core for convex games. In non-convex games it may fall outside; the core may even be empty while the Shapley value still exists. ### Empty core misdiagnosed as "no solution" **Wrong**: "The core is empty, so the game has no solution." **Diagnostic**: Have you considered the nucleolus, kernel, bargaining set, or Shapley value? **Right**: Many solution concepts coexist; an empty core simply rules out *stable* coalitional payoff vectors with the strict-domination property. The nucleolus always exists for games with non-empty imputation set. ## Statistics and inference inside game-theoretic studies ### Treating QRE as a noise model **Wrong**: Adding logit noise to Nash predictions to fit experimental data without re-solving for fixed points. **Diagnostic**: Are your QRE predictions a fixed point of the logit best-response operator, not a perturbation of Nash? **Right**: QRE is solved as a fixed point of the noisy best-response map. The fixed point can be qualitatively different from any "perturbed Nash" because all players' noise interacts. ### Power calculations after the fact **Wrong**: "We didn't reject; let's compute post-hoc power to argue the test was insufficient." **Diagnostic**: Was the power calculation done before data collection, at the smallest effect size of substantive interest? **Right**: Power must be planned ex ante. Post-hoc "observed power" is a deterministic transformation of the p-value and adds no inferential content. ## Reading and reporting ### Equilibrium ≠ prediction **Wrong**: Reporting a model's equilibrium as the *prediction* for real-world behaviour without acknowledging alternatives. **Diagnostic**: Did the equilibrium concept assume best-response or rationality your subjects might not satisfy? **Right**: Distinguish *equilibrium as benchmark* from *equilibrium as point prediction*. Behavioural game theory (QRE, level-$k$, learning) supplies disciplined alternatives. ### Strategy ≠ outcome **Wrong**: "The Nash equilibrium of this game is the payoff vector $(2, 2)$." **Diagnostic**: Did you describe a strategy profile, or just the payoffs at one? **Right**: A Nash equilibrium is a strategy profile $(s_1^*, \dots, s_n^*)$. The payoff vector $(u_1(s^*), \dots, u_n(s^*))$ is the *outcome at* that equilibrium, not the equilibrium itself. ## See also - [Glossary](glossary.qmd) — formal definitions of the terms used above. - [Cheatsheets](../cheatsheets.qmd) — equilibrium concepts at a glance + R idioms. - [Decision Assistant](../decision-tree/decision-assistant.qmd) — interactive picker for the right model + solution concept.