Calculate expected utilities and the probabilities needed for a mixed equilibrium.
By the end: solve a two-action interior mixture without assuming that every random choice is an equilibrium.
Randomize over complete actions
A mixed strategy assigns probabilities to pure strategies. Here p is Row's probability of Left and q is Column's probability of Left. Random draws are independent. The probability of Left, Left is therefore pq. A shared coin could correlate the choices and would define a different joint distribution. In the two matching examples, the players' payoffs sum to zero in every cell. These are zero-sum games. The maintenance and coordination games are not.
Expected utility weights each possible payoff by its probability. For two actions, a mixture cannot beat both constituent actions against a fixed opponent strategy. If one is strictly better, assigning positive probability to the worse one leaves a profitable deviation.
Calculate joint chances before averaging payoffs
A probability describes how much weight a possible action receives. A probability of 0.4 means 40%, with the remaining 60% assigned to the other action. Independence means that learning Row's draw does not change the probability of Column's draw. It lets us multiply the two probabilities to find a cell's chance. These are assumptions about the random choices, not implications of writing a matrix. Two players could each choose Left half the time yet always match if they use the same coin. Their individual percentages would not establish independence.
Use Unequal stakes as a worked calculation. Suppose Row chooses Left with probability 0.4 and Column chooses Left with probability 0.7. Left, Left then has probability 0.4 times 0.7, or 0.28. Left, Right has probability 0.4 times 0.3, or 0.12. Right, Left has probability 0.6 times 0.7, or 0.42. Right, Right receives the remaining 0.18. Adding these four numbers gives 1, which checks that every possible joint action has been counted exactly once. The probability-area picture shows why the products have those sizes.
Independent chances multiply into exact areas
Widths encode Column's chances and heights encode Row's. Every rectangle is drawn to its stated probability scale.
Now weight Row's four utilities by those cell probabilities. The two cells in Row's Left row contribute 0.28 times 3 and 0.12 times negative 3. Their contributions are 0.84 and negative 0.36, totaling 0.48. The Right row contributes 0.42 times negative 2 plus 0.18 times 2, totaling negative 0.48. Adding the two row totals gives expected utility zero. Column receives the negative of each Row payoff in this particular game, so Column's expected utility is also zero. Zero here is an average, not a possible terminal payoff.
There is a shorter calculation that also explains the incentives. Hold Column's 70% Left probability fixed. Row choosing Left for certain would average 0.7 times 3 plus 0.3 times negative 3, which is 1.2. Row choosing Right for certain would average 0.7 times negative 2 plus 0.3 times 2, which is negative 0.8. Row's current mixture weights these conditional averages by 0.4 and 0.6. The result is 0.48 minus 0.48, again zero. Conditional here means calculating while treating Row's own action as already specified.
This calculation exposes a profitable deviation that the zero total hides. Row can replace the current mixture with certain Left and raise expected utility from zero to 1.2. The mixture is therefore not a best response, even though it uses the equilibrium value of p for this game. Equilibrium concerns a pair of strategies, not one correct-looking percentage. To test Row's deviation, retain Column's q at 70%. Changing q as well would compare two joint scenarios rather than the unilateral change required by the Nash test. Column's responses require a separate comparison.
The same zero average leaves Row a profitable deviation
Row choice
Against Column Left, 70%
Against Column Right, 30%
Expected utility
Certain Left
3 × 0.7 = 2.1
−3 × 0.3 = −0.9
1.2
Certain Right
−2 × 0.7 = −1.4
2 × 0.3 = 0.6
−0.8
40% Left, 60% Right
0.7 × (0.4 × 3 − 0.6 × 2) = 0
0.3 × (−0.4 × 3 + 0.6 × 2) = 0
0
Hold q at 70%. Certain Left improves Row's utility by 1.2; changing both probabilities would test something else.
The explorer performs arithmetic over all four cells; it does not draw a random action or generate a history. A probability-area picture likewise represents a distribution, not a record of one hundred trials. In actual repeated draws, observed counts can differ from their expected proportions. Alternating Left and Right deterministically is also different from independent randomization if an opponent can predict the next action. The transferable question is what information the opponent has when choosing. A percentage alone does not specify whether choices are independent, predictable, or linked by a shared random device.
Solve the other player's indifference
For Match or differ, q = one half makes Row's Left and Right equally attractive. Likewise p = one half makes Column indifferent. At that pair of probabilities neither can improve by changing their mix. Indifference alone for Row does not guarantee that Column is also optimizing.
In Unequal stakes, Row's Left payoffs are 3 and −3; Right payoffs are −2 and 2. Column receives the negatives. The equilibrium keeps q = one half but changes p to 0.4. Row's mixture must offset the unequal stakes to make Column indifferent. A probability is determined by the other player's incentive equation.
Use both indifference equations and then check deviations
Indifference means that a player's available alternatives have equal expected utility. It is necessary for using both actions with positive probability as a best response. Otherwise moving weight away from the worse action raises the average. The support of a mixture is the set of actions that receive positive probability. A fully interior mixture puts positive weight on every available action. In this two-action lesson, an interior candidate therefore needs equality between both Row actions and equality between both Column actions. Equalizing only one player's alternatives leaves half of the equilibrium test unfinished.
For Unequal stakes, first solve Row's comparison using Column's q. Left gives 3 with probability q and negative 3 with probability one minus q. Starting at negative 3 when q is zero, each additional unit of q adds 6, giving negative 3 plus 6q. Right starts at 2 and falls by 4q, giving 2 minus 4q. Equality requires the total change of 10q to cover the starting gap of 5. Thus q is 5 divided by 10, or one half. Both Row actions then give zero.
Different players' comparisons determine different probabilities
Both axes use exact linear scales. Row's crossing determines q; Column's crossing determines p.
Column's calculation uses Row's p instead. Against certain Column Left, Column receives negative 3 when Row chooses Left and 2 when Row chooses Right. Its expected utility is 2 minus 5p. Certain Column Right instead gives 3 against Row Left and negative 2 against Row Right, yielding negative 2 plus 5p. The starting gap is 4 and the combined change is 10p. Equality therefore requires p to be 0.4. Substituting gives 2 minus 2 and negative 2 plus 2. Both are zero, completing the second player's indifference check.
The probability belonging to one player comes from the other player's payoff comparison. This does not mean Row wants to help Column become indifferent. If Row chose a larger p, Column would favor Right; a smaller p would favor Left. At an interior equilibrium, neither player can give the opponent a strict preference while expecting that opponent to keep mixing across both actions. Indifference is a consistency condition imposed by mutual best responses. It is not an extra preference for generosity, uncertainty, or equal treatment of the opponent's available actions.
Checking the candidate now needs no search through every possible percentage. At p equal to 0.4 and q equal to 0.5, Row's two pure utilities are both zero, and Column's two are both zero. Any alternative mixture averages zero with zero, so it cannot improve either player's utility. This argument works because the game has only these two actions for each player. In a larger paper game, two equally good actions might both be worse than a third. Equality among supported actions would then be insufficient until that unused alternative was checked too.
Four pure deviations certify the unequal-stakes candidate
Player at p = 40%, q = 50%
Certain Left
Certain Right
Current mixture
Best gain
Row
3 × 0.5 − 3 × 0.5 = 0
−2 × 0.5 + 2 × 0.5 = 0
0
0
Column
−3 × 0.4 + 2 × 0.6 = 0
3 × 0.4 − 2 × 0.6 = 0
0
0
Each current mixture averages two zeros. No alternative mixture can exceed its best pure constituent.
A unique fully interior candidate is narrower than a complete equilibrium classification. The lesson helper returns such a candidate or null. Null can occur because an equation has no interior solution, or because equalities do not determine a unique probability. For a separate paper example, suppose every cell gives both players zero. Every pair of mixtures is a Nash equilibrium, yet there is no unique interior candidate to return. Other games have equilibria that put zero probability on an action. Neither a missing interior candidate nor a solver's limited output proves that all mixed equilibria are absent.
Compare incentives and outcomes
Test both players’ deviation gains
Unequal stakes. Each cell lists Row, then Column utility.
Row / Column
Left
Right
Left
Row: 3 ✓Column: -3
Row: -3Column: 3 ✓
Right
Row: -2Column: 2 ✓
Row: 2 ✓Column: -2
✓ means that player cannot improve by switching alone while the other action stays fixed. Ties receive marks too. A border identifies the selected cell.
Solid: Row LeftDashed: Row Right
Player
Pure Left
Pure Right
Current mixture
Best deviation gain
Row
0
0
0
0
Column
-0.5
0.5
0
0.5
Both deviation gains zero: No. Interior equilibrium: p = 40%, q = 50%.
Weight Row’s two conditional payoffs
Each action’s utility already averages over Column’s q. Use probabilities from zero to one, not raw percentage numbers.
Game changes reset p and q to 50%. Mix presets replace both probabilities; each slider changes only its named player. Draws are independent. The graph varies Column’s probability while holding the payoff table fixed; the table checks both players. No random sample is generated.
Row mix in Unequal stakes
Column Left utility
Column Right utility
p = 0.5
−0.5
0.5
p = 0.4
0
0
Equal randomization is not automatic when stakes differ.
Separate a player's own mixture from the opponent's incentive
Use the controls to isolate two different effects of a probability change. Select Match or differ, then Row biased. This sets Row's Left probability to 80% and Column's to 50%. Row's pure Left and Right utilities are both zero because Column splits evenly. Row's mixture utility is consequently zero as well. Column, however, receives negative 0.6 from certain Left and positive 0.6 from certain Right. The current half-half Column mixture averages these to zero. Column can gain 0.6 by switching to Right, so the pair is not an equilibrium.
Keep q at 50% and change only Row's slider to 50%. Column's pure utilities both become zero and its deviation gain disappears. Row's graph does not change its payoff lines because those lines depend on q, not on Row's chosen p. Now move only q to 75%, retaining p at 50%. Row's Left utility becomes 0.75 minus 0.25, or 0.5, while Right gives negative 0.5. Row's half-half average stays zero, but its deviation gain becomes 0.5. A constant mixture payoff can therefore conceal a changing incentive problem.
Equal mixture utilities do not imply equal deviation gains
p, q
Row Left / Right
Column Left / Right
Mixture utilities Row, Column
Gains Row, Column
80%, 50%
0 / 0
−0.6 / 0.6
0, 0
0, 0.6
50%, 50%
0 / 0
0 / 0
0, 0
0, 0
50%, 75%
0.5 / −0.5
0 / 0
0, 0
0.5, 0
These Match or differ settings retain one opponent probability while changing the other. Gains compare unilateral alternatives.
Return to Equilibrium mix before switching to Unequal stakes. A game change resets both sliders to 50%, rather than preserving the previous equilibrium probabilities as a promise that they remain correct. At that reset pair, Row is indifferent with utility zero. Column's pure Left gives negative 0.5 and pure Right gives 0.5, so Column can improve by 0.5. Selecting Equilibrium mix now replaces both probabilities with 40% and 50%. Only this pair removes both gains. The named preset is a shortcut to a checked calculation, not evidence that every nearby mixture is equivalent.
For a more diagnostic perturbation, retain Unequal stakes and set p to 40%, q to 70%. Row's conditional utilities become 1.2 and negative 0.8, while Column remains indifferent at zero. Row's gain is 1.2 even though both current mixture utilities remain zero. Now change only p to 100%. Row takes the best response and earns 1.2, but Column's pure Right would give 3 instead of Left's negative 3. Column's current 70% Left mixture gives negative 1.2 and leaves a gain of 4.2. One player's improvement need not solve the game for both.
The graph and table answer related but different questions. The graph displays Row's expected utility from each pure action as Column's probability varies across the horizontal axis. Its crossing identifies the q that makes Row indifferent. It does not identify p by itself. The table uses the currently selected pair and reports each player's pure alternatives, actual mixture utility, and best unilateral gain. Read the gain as the best pure utility minus current mixture utility. It is never a social loss measure and does not compare utilities across the two people.
A best response for Row can leave Column further from optimal
p
Row Left / Right
Column Left / Right
Current utilities Row, Column
Gains Row, Column
40%
1.2 / −0.8
0 / 0
0, 0
1.2, 0
100%
1.2 / −0.8
−3 / 3
1.2, −1.2
0, 4.2
Unequal stakes, q fixed at 70%. Row improves by choosing Left, but Column then strongly prefers Right.
These experiments are comparative calculations, not an adjustment process implemented by the lab. Moving Row to its best response and then Column to its best response can illustrate why predictability is exploitable, but no control instructs the players to learn or converge. A visible sequence of your slider choices is not a simulated equilibrium-selection rule. Transfer the method by specifying what stays fixed in each comparison, calculating the conditional alternatives first, and then checking both players' gains at the same pair. That procedure distinguishes an optimal response to one forecast from mutual optimization.
Interpret probabilities carefully
The explorer reports exact expected utilities up to displayed rounding, not a short-run win guarantee. A literal randomizer, an observer's uncertainty about an action, and a population fraction are different interpretations of mixing. They need not imply the same behavior over time.
Pure best-response reasoning survives any strictly increasing transformation of a player's payoffs. Expected-utility mixtures generally do not. To preserve expected-utility comparisons, use a positive affine transformation of that player's utility scale. Do not replace utility with arbitrary ordinal ranks before calculating averages.
Check what the probability and utility scales mean
Independent half-half choices in Match or differ assign 25% to each cell. Row matches half the time and differs half the time, so its expected utility is zero. Contrast a separate paper arrangement where a public coin tells both players the same action. Each player still chooses Left half the time, but only Left, Left and Right, Right occur, each with probability 50%. Row now always receives 1 and Column negative 1. Equal individual percentages can describe different joint outcomes. The independent-mixture calculation cannot be reused without checking the relationship between draws.
That public-coin arrangement is not automatically an equilibrium under voluntary obedience. After seeing a recommendation to choose Left, Column knows Row will choose Left if Row obeys. Column would then gain by choosing Right instead. A recommendation tied to a shared signal introduces conditional deviation checks, which the current independent-mixture lab does not solve. Correlation means that one draw carries information about another; it does not mean recommendations enforce themselves. In applications, distinguish a common random signal, an enforceable joint choice, and separate private randomizers before using their probabilities to predict payoffs.
The same individual percentages can produce different joint chances
Paper comparison only. Shared recommendations create conditional incentives; Column would reject the instruction to match.
Utility scales also matter when taking averages. A positive affine transformation multiplies every utility of one player by the same positive number and then adds the same constant. For example, replace each Row utility u with twice u plus 5. Expected utility then also doubles and increases by 5 because the probability weights sum to 1. Every difference between Row's alternatives doubles, preserving its sign and all ties. In Unequal stakes the transformed Row entries are 11, negative 1, 1, and 9. At q equal to one half, both pure averages become 5.
An arbitrary increasing transformation preserves rankings of certain outcomes but can change lottery comparisons. Consider a separate paper choice between a sure utility of 2 and an equal-chance lottery giving 0 or 3. The original expected utilities are 2 and 1.5, favoring certainty. Cubing these utility numbers preserves their order but changes the averages to 8 and 13.5, favoring the lottery. Cubing is increasing even across negative numbers, so the failure is not caused by reversing any certain-outcome ranking. It changes the numerical tradeoff between possible outcomes when uncertainty is involved.
This is why recording only first, second, third, and fourth preferences is insufficient to justify mixed-strategy arithmetic. Expected-utility numbers also encode preferences about uncertain prospects under the model's assumptions. They are not automatically cash, happiness measurements, or interpersonal welfare units. A common rescaling within Row's utility system does not authorize adding Row's result to Column's and interpreting the sum as social benefit. Zero-sum likewise describes the specified payoff representation. It is an assumption of these matching examples, not a general property of strategic interaction or a claim about every consequence outside the model.
Rescaling utility and changing lottery preferences are different operations
Utility rule
Sure outcome originally 2
Equal-chance lottery originally 0 or 3
Preferred
Original utility
2
0.5 × 0 + 0.5 × 3 = 1.5
Sure
Twice utility plus 5
9
0.5 × 5 + 0.5 × 11 = 8
Sure
Utility cubed
8
0.5 × 0 + 0.5 × 27 = 13.5
Lottery
Paper choices: positive affine rescaling preserves the ranking; cubing preserves certain-outcome order but reverses this lottery comparison.
Finally, keep the interpretation of a probability separate from its algebra. A literal private randomizer assigns chances to your own draw. A belief describes your uncertainty about another person's action. A population proportion describes how often different people use an action. These can sometimes produce the same expected-utility formula while requiring different evidence and information assumptions. The linked Yale lecture on interpreting mixed strategies distinguishes these interpretations; they do not establish which describes any particular learner or institution. The lab checks stipulated independent mixtures and deviations. Establishing actual randomization or a behavioral frequency would require observations beyond this calculation.
Exercises
Check a one-sided mixture
In Match or differ, Row uses p = 0.8 and Column uses q = 0.5. Row is indifferent. Is this a Nash equilibrium?
Hint
Check Column's two pure expected utilities.
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.
Column action
Expected utility
Left
−0.6
Right
0.6
Current half-half mix
0
One player's indifference does not settle both incentives.
No. Column's Left utility is −0.8 + 0.2 = −0.6; Right gives 0.8 − 0.2 = 0.6. Their current half-half mixture gives 0, so moving entirely to Right gains 0.6. Both players must be best responding.
Solve the unequal mixture
In Unequal stakes, find the p that makes Column indifferent. Explain why q remains one half.
Hint
Column's Left utility is 2 − 5p and Right is 5p − 2.
Worked answer
Unequal stakes. Each cell lists Row, then Column utility.
Row / Column
Left
Right
Left
Row: 3 ✓Column: -3
Row: -3Column: 3 ✓
Right
Row: -2Column: 2 ✓
Row: 2 ✓Column: -2
✓ means that player cannot improve by switching alone while the other action stays fixed. Ties receive marks too. A border identifies the selected cell.
Indifference check
Substitution
Utility
Column Left
2 − 5 × 0.4
0
Column Right
5 × 0.4 − 2
0
Row Left
6 × 0.5 − 3
0
Row Right
2 − 4 × 0.5
0
The two equations determine different players' probabilities.
Equating Column's utilities gives 2 − 5p = 5p − 2, hence 10p = 4 and p = 0.4. For Row, Left gives 6q − 3 and Right gives 2 − 4q. Equating them gives 10q = 5 and q = 0.5. Both probabilities are strictly between zero and one, and there are no other actions to check.
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.
Selected expected-payoff and indifference derivations. Supported actions must be best responses, not merely equal to each other if other actions exist.