Stability, efficiency and solution assumptions

What you will learn

Distinguish a response to fixed beliefs, mutual unilateral stability and pure-profile Pareto efficiency, including ties, multiple equilibria and the absence of pure equilibria.

Prerequisites: Complete plans and information sets · Solving finite chip games backward · Mixed strategies · Exploitability and model limits

A good action needs a stated comparison

A best response maximizes one player’s expected payoff against a specified opponent policy. This is a conditional claim. If Row optimizes against a fixed Column policy, nothing in that calculation establishes that Column is also best responding to Row. A mutually consistent profile requires checking both players. A robust guarantee instead asks what can be secured across allowed opponent responses; a high receipt against a friendly policy is not such a guarantee.

Our activity fixes a known two-player simultaneous game. Row chooses AorB; Column chooses LorRwithout seeing the other player’s current choice. Each cell lists a pair of linear fake-chip receipts(Row,Column). Classroom counterfactual inspection reveals what would happen under a different choice; it does not grant an in-game player advance observation. A unilateral deviation changes one action while the other action stays fixed. The other player’s receipt can still change.

Pure equilibrium tests unilateral incentives, not joint welfare

A pure Nash profile has no strictly profitable unilateral deviation for either player. For a2×2game, check Row’s alternative row while holding Column fixed, then Column’s alternative column while holding Row fixed. A tie is allowed: moving for the same payoff is not a strict improvement. The property concerns the specified incentives; it does not say that actual people must choose that profile or that the profile is unique.

In the first original table, receipt pairs A/L(4,4), A/R(0,6), B/L(6,0), B/R(1,1)make B/Rthe only pure equilibrium. Against L, Row prefers6to4; against R, Row prefers1to0. Column has the symmetric incentive for R. At B/Rneither benefits from switching alone. Yet both could receive4at A/L. Stability does not rule out a coordinated improvement, because the unilateral test never changes both actions together.

Pareto comparisons preserve each player’s direction

Among the four listed pure profiles, an alternative Pareto dominates a profile when neither player receives less and at least one receives strictly more. A profile is efficient within that listed set if no such alternative exists. Identical payoff pairs do not strictly dominate one another. This is a restricted pure-profile frontier, not an assertion about all mixed or correlated lotteries.

For the first table, A/L, A/Rand B/Lare all on that frontier. The larger total8at A/Ldoes not Pareto dominate A/R: Column would lose6−4=2. B/Ris dominated by A/Land is not on the frontier despite being the sole pure equilibrium. Maximizing a sum is yet another criterion; here the sum has the clear unit of fake-chip receipts. For arbitrary utility indices, adding across players needs additional interpersonal comparability assumptions.

The solution set can be multiple or have no pure member

In the second table A/L(5,5), A/R(2,0), B/L(0,2), B/R(2,2), both A/Land B/Rare pure equilibria. At B/R, each unilateral switch ties the switching player’s receipt2, so neither strictly benefits. A/Ldominates every other listed profile and is the only pure-profile efficient cell. An equilibrium label alone neither selects A/Lnor excludes the inferior B/R. Refinement, communication, enforcement or learning dynamics would introduce extra claims that need their own assumptions.

In the third table, A/Land B/Rpay(3,0), while A/Rand B/Lpay(0,3). At every pure cell one player can switch alone and improve from0to3, so there is no pure equilibrium. All four cells are efficient among the listed pure profiles: improving one owner from0to3lowers the other from3to0. No pure equilibrium is not no equilibrium at all. Independent half-half mixing makes each opponent action indifferent and gives mean receipt1.5to each in this particular table. The activity does not search every mixed profile, predict human convergence or solve full poker.

Hands-on: Test deviations and separate stability from efficiency

Probe all unilateral switches, mark strict own-receipt improvements, construct complete pure equilibrium and Pareto sets and audit their different claims across three games.

Reading solution workshop…

Common mistakes

• Holding the other payoff fixed rather than the other action.
• Treating a tie as a strictly profitable deviation.
• Calling the largest joint receipt a Nash condition.
• Claiming that no pure equilibrium means no mixed equilibrium.
• Selecting only the equilibrium one prefers while omitting another valid profile.
• Calling a pure-profile frontier globally efficient over all possible lotteries.
• Turning an equilibrium calculation into a guarantee of actual human behavior.

Keep these ideas

• Best response, robust guarantee, equilibrium and welfare use different comparisons.
• A pure equilibrium forbids strictly profitable unilateral deviations for every player.
• A Pareto improvement can change both actions and must not harm either player.
• Multiple or absent pure equilibria require honest set-valued or restricted conclusions.

Further reading

• Ben Polak — ECON159 Lecture5, Nash equilibrium and coordination (Open Yale Courses) ↗

  Bounded official transcript review of Chapter1and3, plus the Pareto/coordination distinction in Chapter4. Original receipt tables, fake-chip ledgers and artwork; not full-video review, human-play prediction or a full-poker solution. Checked 2026-10-22.