Check Nash equilibrium one deviation at a time, then ask what it does not prove.
By the end: identify pure equilibria and distinguish stability against unilateral deviation from efficiency.
Each player checks their own coordinate
A Nash equilibrium is a strategy profile where no player can improve their payoff by changing only their own strategy. In a pure strategy, a player selects one action with certainty. Mark Row's largest entry within each column and Column's largest entry within each row. Cells with both marks are pure Nash equilibria.
Mutual Skip is the maintenance game's only pure equilibrium. Changing alone would lower the deviator's payoff from 1 to 0. Mutual Maintain is not an equilibrium: either player can raise their own payoff from 3 to 5 by skipping. The test holds the other strategy fixed; it does not assume the other player retaliates within this one-shot game.
Test a proposal without changing the opponent's choice
Treat an alleged equilibrium as a proposal containing one action for each player. Give each player a separate opportunity to revise their own part while the other part stays fixed. In Shared maintenance, the proposal Maintain, Skip gives Row zero. Holding Column at Skip, Row can choose Maintain for zero or Skip for one. The gain from switching is one minus zero, which is one. That single positive gain rejects the proposal.
Now test Skip, Skip without assuming either neighbor is pleased with the pump. Row receives one and would receive zero by maintaining alone. Column faces the same comparison. Both possible unilateral changes reduce the changer's utility by one, so neither change is profitable. Equilibrium describes these comparisons, not the physical quality of the outcome. It also does not require a conversation, a written agreement, or affection between the players. Each selected action only needs to be a best response to the other selected action under the stipulated preferences.
Only one coordinate changes in each test
Arrows are unilateral alternatives from Maintain, Skip. Row can gain one; Column would lose two. The diagonal joint change is not either player's unilateral option.
The word deviation means a change in one player's strategy relative to the proposed profile. It does not mean cheating or breaking a legal obligation. When checking a deviation here, the other neighbor does not immediately respond. A response to observed skipping would require additional timing and information, or a repeated interaction with a future round. Those are different games taught later. Letting Column retaliate inside this calculation would silently replace the present one-shot choice with a longer interaction and could reverse the comparison for reasons absent from this matrix.
A useful rejection statement names the player, the held-fixed opponent action, and the two own utilities. For example, at Maintain, Maintain, Row receives three and can obtain five by skipping against Maintain. The gain is two. Column has the same opportunity, but either one suffices to disprove equilibrium. By contrast, saying that a different cell has larger numbers does not identify a feasible unilateral move. Reaching Skip, Maintain from Maintain, Skip changes both coordinates, so it cannot serve as Row's deviation in the test of that original proposal.
Best-response marks summarize comparisons rather than replace them. Row's marks compare the first numbers vertically within one column. Column's marks compare the second numbers horizontally within one row. You never compare Row's three with Column's five to decide who should switch. Each utility scale represents that player's preferences. Even if both happen to use small integers, subtracting one person's utility from another's has no established meaning here. Subtracting Row's own stay utility from Row's own alternative utility does have a meaning in this stipulated numerical representation.
A positive switching gain rejects the proposal
Proposed profile
Row gain
Column gain
Nash?
Maintain, Maintain
5 − 3 = 2
5 − 3 = 2
No
Maintain, Skip
1 − 0 = 1
3 − 5 = −2
No
Skip, Maintain
3 − 5 = −2
1 − 0 = 1
No
Skip, Skip
0 − 1 = −1
0 − 1 = −1
Yes
Each gain subtracts that player's stay utility from their switch utility, holding the opponent fixed. Only mutual Skip has no positive gain.
For a two-action game, each player has only one alternative pure action at a selected cell, so these two checks exhaust the pure deviations. A larger game would require checking every alternative, not just a nearby or familiar action. The scope matters when someone reports that a proposal survived testing. A calculation that examined one possible change cannot certify stability against an omitted third action. The current explorer deliberately has exactly two players and two actions each.
Efficiency is a different test
An outcome Pareto-dominates another if it makes at least one player better off and none worse off. Mutual Maintain Pareto-dominates mutual Skip because both receive 3 rather than 1. An outcome is Pareto efficient within a specified feasible set if no feasible alternative Pareto-dominates it.
Equilibrium checks unilateral incentives; Pareto efficiency compares feasible joint outcomes. Neither guarantees equality or fairness. The explorer's Pareto label compares only the four pure cells, not lotteries or outside agreements. Changing the feasible set can change the efficiency claim.
Compare outcomes on two separate utility axes
For a Pareto comparison, stop asking who can move alone. List the feasible joint outcomes and compare each player's ranking of them. In Shared maintenance the utility pairs are three, three; zero, five; five, zero; and one, one. Moving from one, one to three, three raises both utilities by two. Moving from one, one to five, zero raises Row's utility but lowers Column's. The first comparison is a Pareto improvement; the second is a conflict that this criterion cannot resolve. A larger number for one person is insufficient.
Plot Row utility horizontally and Column utility vertically. Points to the right help Row; points above help Column. Relative to one, one, a point at least as far right and at least as high helps neither player less. At least one coordinate must improve strictly to establish Pareto dominance. The point three, three meets that test. The point five, zero does not because it lies below. This geometry uses two separate utility axes. It does not turn their units into a common welfare scale or justify adding the coordinates.
Joint improvements lie above and to the right
Axes use separate stipulated utilities, with one grid interval per unit. Only three, three Pareto-dominates one, one among these four pure maintenance outcomes.
Within these four pure cells, three, three is Pareto efficient. Neither unequal outcome improves both coordinates, and one, one is worse for both. The unequal pairs zero, five and five, zero are also efficient within this set. Column cannot retain five while Row improves on zero at another displayed pure cell; the symmetric statement holds at five, zero. Thus three efficient outcomes coexist with only one pure Nash equilibrium, which is inefficient. Efficiency neither selects one attractive distribution nor guarantees that an individual can bring about the proposed improvement.
The phrase within this set is part of the claim. Consider a separate paper example, not an explorer preset, with feasible pure utility pairs four, zero; zero, four; one, one; and zero, zero. Among these points, one, one is efficient because each other point lowers at least one coordinate. Now suppose a fair public lottery can choose between four, zero and zero, four. Its expected utilities are two and two. Under expected-utility evaluation, adding that lottery to the feasible set makes one, one inefficient. The available alternatives changed; the original comparison was not miscalculated.
A public lottery over joint outcomes is not the same object as two people independently randomizing their actions. The lottery in the paper example selects exactly one of two specified pairs, with equal chances. It never selects the other pairs. Independent private choices generally produce a different distribution. Later lessons calculate those distributions explicitly. For now, the point is narrower. An efficiency label needs a named feasible set and an evaluation rule for uncertain outcomes. The explorer uses only its four pure cells, so it does not perform this expanded lottery comparison.
A new feasible lottery changes an efficiency judgment
Feasible alternative
Row expected utility
Column expected utility
Pure outcome A
4
0
Pure outcome B
0
4
Pure outcome C
1
1
Pure outcome D
0
0
Half A, half B
½ × 4 + ½ × 0 = 2
½ × 0 + ½ × 4 = 2
Separate paper example. Each lottery selects a complete joint outcome, not independent player actions. Its expected pair dominates one, one when lotteries are feasible.
Fairness asks additional questions that Pareto efficiency leaves unanswered. A person at zero in an efficient pair might reasonably object to the distribution. Calling the pair efficient only says that the specified alternatives contain no improvement for someone without a loss for someone else. It does not establish consent, adequate living conditions, or a defensible entitlement. Conversely, a joint improvement need not explain how to secure agreement or prevent later deviations. Keep the evaluation of outcomes separate from the incentives needed to implement them and the ethical reasons for preferring one distribution.
Compare incentives and outcomes
Inspect one action change at a time
Shared maintenance. Each cell lists Row, then Column utility.
Row / Column
Maintain
Skip
Maintain
Row: 3Column: 3
Row: 0Column: 5 ✓
Skip
Row: 5 ✓Column: 0
Row: 1 ✓Column: 1 ✓Both best responses
✓ means that player cannot improve by switching alone while the other action stays fixed. Ties receive marks too. A border identifies the selected cell.
Selected payoffs: 3, 3. Nash equilibrium: No. Pareto efficient among pure cells: Yes.
Unilateral comparison
Stay
Switch alone
Row; Column holds Maintain
3
5
Column; Row holds Maintain
3
5
Player
Strictly dominant action
Weakly dominant action
Row
Skip
Skip
Column
Skip
Skip
Under our definition, strictly dominant actions also satisfy weak dominance.
A game preset replaces the payoffs and resets both actions to the first listed choice. Action buttons change only that player’s choice. Utilities are stipulated. Pareto efficiency below compares only the four pure cells, not lotteries or outside options.
Shared maintenance. Each cell lists Row, then Column utility.
Row / Column
Maintain
Skip
Maintain
Row: 3Column: 3
Row: 0Column: 5 ✓
Skip
Row: 5 ✓Column: 0
Row: 1 ✓Column: 1 ✓Both best responses
✓ means that player cannot improve by switching alone while the other action stays fixed. Ties receive marks too. A border identifies the selected cell.
Profile
Unilateral incentive
Pareto comparison
Maintain, maintain
Either can gain by skipping
Both prefer it to mutual Skip
Skip, skip
Neither can improve alone
Dominated by mutual Maintain
Nash and Pareto answer different questions.
Use the explorer to distinguish ties from profitable changes
Select Shared maintenance in the explorer, then select Row Maintain and Column Skip. The selected border identifies the proposal, while check marks elsewhere identify conditional best responses. Read the unilateral comparison table before reading the Nash status. Row's stay and switch values are zero and one. Change only Row to Skip and verify that Column remains at Skip. Row's values now read one and zero. This controlled comparison shows why the second cell passes where the first fails. No estimate of how often neighbors maintain enters either calculation or the status label.
Keep Skip, Skip selected and compare the Pareto status with the Nash status. They disagree because their questions differ, not because one calculation overrides the other. Then change to Joint project. A game preset resets both selected actions to their first listed choices, so the selected profile becomes Join, Join. Both receive six, and either unilateral switch gives four. Select Safe for both players and repeat the check. Both receive four, and a lone switch to Join gives zero. This game has two pure equilibria even though neither action dominates the other.
Bottom, Left survives both deviation checks
Row / Column
Left
Right
Top
2, 2
1, 1
Bottom
2, 0 ★
0, 0
At ★
Stay
Switch
Row
2
2
Column
0
0
Actual Ties matter utilities. A zero gain leaves a best response. Top's strict advantage against Right does not create a gain against the selected Left.
Next select Ties matter and set Row to Bottom while leaving Column at Left. Row gets two; switching to Top also gives two. Column gets zero; switching to Right also gives zero. Both players therefore receive best-response marks at Bottom, Left. Nash equilibrium rules out a strictly profitable deviation, not an equally good alternative. A report that rejects this profile merely because Row could switch confuses availability with improvement. The same mistake would reject any equilibrium where a player has two equally attractive actions against the selected opponent choice.
Inspect the dominance readout in that same game. Top weakly dominates Bottom because it ties at two against Left and gives one rather than zero against Right. It does not strictly dominate Bottom. There is no contradiction between Bottom being weakly dominated and Bottom, Left being a Nash equilibrium. The strict advantage of Top occurs against Right, which is not Column's selected action in that profile. Dominance examines both possible opponent actions; this equilibrium test examines the particular opponent action in the proposal. A distinction from the previous chapter now explains an equilibrium rather than merely naming it.
Finish by selecting Match or differ. At Left, Left the utilities are one and negative one. Column switches to Right to obtain one instead. At Left, Right, Row can switch to Right for one instead of negative one. Continue through Right, Right and Right, Left and identify the corresponding dissatisfied player. Every cell fails, but for different unilateral comparisons. The absence of a pure Nash profile is an output of checking all four proposals. It does not follow from low utility, unequal utility, or the fact that the two players have opposing preferences.
Every pure matching profile has a profitable deviation
These arrows enumerate deviations, not a simulated time path. Each switch raises the deviator's utility from negative one to one while leaving the other action fixed.
These explorations change selected actions or replace a preset game; they do not edit individual payoffs, delete dominated strategies, or simulate learning. The border is your selection, not a predicted outcome, and all best-response marks remain visible even in unselected cells. Record the actual reason a status changes. In one case you altered Row while retaining Column; in another you replaced every payoff through a preset. Keeping those interventions distinct prevents a common mistake in strategic reasoning: attributing a changed conclusion to an action choice when the underlying preferences or available game also changed.
Existence does not supply a forecast
Some games have several pure equilibria and some have none. Match or differ has no pure equilibrium: Row wants matching choices and Column wants different choices. For any fixed cell, someone wants to switch. This does not mean the game is inconsistent.
Every finite normal-form game has a Nash equilibrium when independent mixed strategies and expected-utility evaluation are allowed. The theorem supplies existence, not uniqueness, a convergence process, accurate beliefs, or observed behavior. The next lessons develop coordination and randomization rather than assuming a marked cell will occur.
Separate equilibrium checks from a process that reaches equilibrium
A list of mutual best responses does not describe a learning procedure. To see the difference, imagine a paper adjustment rule in Joint project. Each player observes yesterday's opponent action and chooses a best response to that old action today. Start yesterday at Join, Safe. Row sees Safe and switches to Safe; Column sees Join and switches to Join. The next pair is Safe, Join. Repeating the same simultaneous update returns to Join, Safe. The two players keep missing each other despite two pure equilibria being available in the unchanged game.
This constructed cycle is not generated by the explorer and is not a claim about typical human learning. It isolates a missing assumption in statements such as rational adjustment must reach equilibrium. If only one player updates at a time, the same starting pair can instead reach Safe, Safe when Row updates first, or Join, Join when Column updates first. Update timing changes the process while leaving the underlying static best-response calculations unchanged. An equilibrium theorem cannot choose between these adjustment rules, nor can the marked cells supply evidence that people use either rule.
The update rule can change the outcome
From Join, Safe
First update
Next update
Result
Both reply to old actions
Safe, Join
Join, Safe
Cycle repeats
Only Row updates first
Safe, Safe
Safe, Safe
Lower equilibrium
Only Column updates first
Join, Join
Join, Join
Higher equilibrium
Paper adjustment processes, not explorer playback. Simultaneous replies to old actions cycle; a single player's first update instead reaches one of the existing pure equilibria.
The absence of a pure equilibrium in Match or differ requires another extension. A mixed strategy specifies probabilities for choosing pure actions. A probability of one half means each action receives half the chance, not that a player chooses half of an action. At independent half-half choices, each of the four cells has probability one quarter. Row's expected utility is one quarter plus negative one quarter plus negative one quarter plus one quarter, which is zero. Column's is also zero. Expected utility is a probability-weighted average of that player's possible utility values.
Against an opponent who uses half-half, either pure action gives expected utility zero. Any mixture of those pure actions also gives zero, so neither player can improve by changing their own probability. This supplies a mixed equilibrium even though no realized pure cell passes the pure equilibrium test. The equilibrium object is the pair of probability rules. A single realized outcome still has a player who would prefer a different action if the opponent's realized action were known in advance. That hindsight comparison is not the ex ante comparison made before the independent draws.
The finite-game existence result permits these probability rules and assumes expected-utility evaluation. It does not say that a unique fully interior solution always exists. Fully interior means every available pure action receives positive probability. Some games have only pure equilibria; some have several equilibria or ties that allow many mixtures. The later mixing explorer solves two specific examples rather than every possible game. Here the theorem's role is to explain why failing to find a pure profile is not equivalent to proving that no Nash equilibrium exists under the broader strategy definition.
A mixed equilibrium is a distribution, not a pure cell
Equal rectangle areas encode independent half-half choices. Opposite signs cancel in Row's expected utility. No sampled outcomes or frequencies are generated by this figure.
When transferring this method to a new claim, specify the proposal and the strategy set before testing it. Ask whether the proposed object is a pure action pair, a pair of independent probability rules, or a plan in a longer interaction. Then check each player's feasible deviations under the same information assumptions. Only after that ask which outcomes are efficient, how players might coordinate on one, and what observations support that prediction. A correct incentive check is useful precisely because its conclusion is limited. It can reject a proposed equilibrium without pretending to explain every possible observed choice.
Exercises
Audit a proposed equilibrium
A report calls Maintain, Skip an equilibrium because Column earns the maximum payoff 5. What check is missing?
Hint
Inspect Row's available deviation.
Worked answer
Shared maintenance. Each cell lists Row, then Column utility.
Row / Column
Maintain
Skip
Maintain
Row: 3Column: 3
Row: 0Column: 5 ✓
Skip
Row: 5 ✓Column: 0
Row: 1 ✓Column: 1 ✓Both best responses
✓ means that player cannot improve by switching alone while the other action stays fixed. Ties receive marks too. A border identifies the selected cell.
Row action
Row utility
Maintain against Skip
0
Skip against Skip
1
The missing deviation increases Row's payoff.
Row earns 0 and can switch to Skip to earn 1 while Column stays at Skip. One profitable unilateral deviation is enough to reject equilibrium. Column's maximum payoff does not remove Row's incentive.
No pure equilibrium
For Match or differ, examine Left, Left and Left, Right. Who wants to switch?
Hint
Read Row's match preference and Column's mismatch preference.
Worked answer
Match or differ. Each cell lists Row, then Column utility.
Row / Column
Left
Right
Left
Row: 1 ✓Column: -1
Row: -1Column: 1 ✓
Right
Row: -1Column: 1 ✓
Row: 1 ✓Column: -1
✓ means that player cannot improve by switching alone while the other action stays fixed. Ties receive marks too. A border identifies the selected cell.
Current cell
Deviator
Their utility change
Left, Left
Column
−1 to 1
Left, Right
Row
−1 to 1
A deviation need not improve the other player's payoff.
At Left, Left, Column can switch to Right and raise their utility from −1 to 1. At Left, Right, Row can switch to Right and raise their utility from −1 to 1. The other two cells have symmetric deviations. There is no pure equilibrium; an independent half-half mix will later solve both incentives.
Sources and further reading
Selected reference passages support the definitions and algorithms below. The games, explanations, and implementations are original. Terminology and assumptions are stated within each lesson; source review is not an independent audit of this course. Optional links are not runtime dependencies.