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Mathematics in Bukit Timah | Game Theory, Payoff Matrices and Nash Equilibrium

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by

eduKate Asia
Illustration of two students discussing mathematical reasoning with square tiles and a notebook.

Did you know that a decision can be individually sensible even when everyone would prefer a different outcome? In a fictional two-person points game, both players can receive three points by choosing the same cooperative action. Yet if each player can gain five points by changing their own action while the other cooperates, the cooperative outcome may be unstable. This is a central puzzle in game theory: the difference between a good joint outcome and a Nash equilibrium.

This Mathematics in Bukit Timah guide explains payoff matrices, strategic choices, best responses, dominant strategies, the Prisoner’s Dilemma, coordination games, pure and mixed strategies, zero-sum games, repeated interaction, Pareto comparisons and backward induction. Every player, classroom decision, token and point score below is fictional. It is not an account of actual Bukit Timah parents, pupils, tutor decisions, school policies or local businesses. We use small exact tables to understand the mathematical logic of interaction.

Quick answer: A Nash equilibrium is a profile of strategies where no single player can improve their own payoff by changing only their own strategy while the others’ strategies stay fixed. It does not necessarily maximise the sum of payoffs, provide fairness or describe what people will actually do. In the classic illustrative payoff matrix below, both selecting D is the unique pure-strategy Nash equilibrium, even though both selecting C would give each more points.

Article contents

Original education illustration with pupils comparing mathematical examples and counterexamples; all strategic payoff examples here are fictional.

What makes a situation a game?

In mathematical game theory, a game specifies players, available strategies and the payoff each player receives for every relevant combination of choices.

A payoff is a numerical representation of the objective defined by the model. It might be an abstract utility number or a concrete score, depending on the setup.

Without clearly stated actions and payoffs, the phrase “winning strategy” is ambiguous. Different objectives can lead to different mathematical predictions.

Why a Bukit Timah classroom is only a fictional setting

Families near Beauty World, Sixth Avenue and King Albert Park may recognise puzzles where two people select actions independently.

Our named players A and B represent abstract decision-makers. Their payoff numbers are invented for teaching and do not describe real pupils’ motivations, relationships or ethical behaviour.

A mathematical model of self-interested utility does not establish how any actual child, parent or organisation should behave.

Begin with a two-player decision table

Suppose players A and B independently choose either C (cooperate) or D (defect).

The payoff matrix gives an ordered pair (A’s payoff, B’s payoff) for each outcome.

The first entry always belongs to A and the second to B. Swapping these entries would change the game unless the whole player labelling were changed consistently.

The fictional Prisoner’s Dilemma payoff table

Use these four outcomes:

A’s actionB chooses CB chooses D
A chooses C(3,3)(0,5)
A chooses D(5,0)(1,1)

The numbers are fictional utility points. They are not a real punishment schedule, real test marking rubric or tuition centre rule.

Each player prefers more of their own points within the stated model.

Work through one cell before comparing strategies

When A chooses C and B chooses D, the payoff pair is (0,5).

A receives zero points and B receives five. The pair (0,5) is not a coordinate on a school map or a single two-dimensional score with magnitude five.

Reading payoff order correctly is essential when identifying which action benefits which player.

Worked example 1: a best response when B cooperates

Hold B’s action fixed at C.

If A chooses C, A receives three. If A chooses D, A receives five.

Because five exceeds three, A’s best response to B choosing C is D. Only A’s payoff is relevant for A’s best response.

A best response depends on the other player’s action

A best response is a strategy giving a player at least as high a payoff as every available alternative, given the other players’ specified strategies.

It does not necessarily maximise the two players’ combined points.

To identify a best response, fix the opponent’s choice and compare one player’s own numbers across their alternative actions.

Worked example 2: a best response when B defects

Now hold B’s action fixed at D.

If A chooses C, A gets zero. If A chooses D, A gets one.

A again prefers D. Thus A chooses D as best response whether B chooses C or D in this fictional game.

A dominant strategy beats alternatives under every opponent choice

A strategy is strictly dominant when it provides a strictly higher payoff than each alternative against every possible opponent strategy.

For player A, D gives 5 instead of 3 when B chooses C, and 1 instead of 0 when B chooses D.

Therefore D strictly dominates C for A. The same comparison is symmetric for B.

Worked example 3: B’s dominant action

Hold A choosing C. B’s own payoff is three from C and five from D, so B prefers D.

Hold A choosing D. B gets zero from C and one from D, so B again prefers D.

Thus D is strictly dominant for B. Both players have the same dominant-strategy structure.

What is a pure-strategy Nash equilibrium?

A pure strategy specifies one definite action for each player.

A pure-strategy Nash equilibrium is a strategy profile in which no player can gain by changing only their own chosen action.

The test is local to unilateral deviations: keep every other player’s action fixed while comparing one player’s alternatives.

Worked example 4: test the D,D outcome

At (D,D), payoffs are (1,1).

If A alone switches to C while B remains at D, A’s payoff falls from one to zero. If B alone switches to C while A remains at D, B’s payoff also falls from one to zero.

Neither player can improve by changing alone. Therefore (D,D) is a Nash equilibrium.

Why the cooperative outcome is not stable here

At (C,C), payoffs are (3,3), which are higher for both players than (1,1).

But if B remains at C and A switches to D, A’s payoff increases from three to five. B has a symmetric incentive.

Thus (C,C) fails the Nash equilibrium test, even though it gives the two players a better joint result under these payoffs.

Worked example 5: compare welfare and stability

At (C,C), total points are 3+3=6.

At (D,D), total points are 1+1=2.

The first has the larger sum, but the second is the stable pure-strategy Nash profile. Maximum total payoff and individual unilateral stability answer different questions.

A Nash equilibrium is not automatically a moral recommendation

The word defect is part of the standard game-theory label, not a judgement about a real person.

A payoff table may omit trust, identity, institutional rules, long-term relationships and social values. Actual behaviour can also deviate from narrow utility-maximising assumptions.

The mathematical conclusion is conditional: given these stated payoffs and this one-shot strategic model, D is strictly dominant. It does not instruct real students to behave selfishly.

Pareto improvement describes a separate comparison

One outcome Pareto-dominates another when at least one player is better off and no player is worse off.

In the fictional Prisoner’s Dilemma, (C,C) with (3,3) Pareto-dominates (D,D) with (1,1).

A Nash equilibrium may therefore be Pareto-inferior to another feasible outcome. Game theory distinguishes stability from the collective welfare comparison.

Worked example 6: another Pareto comparison

Compare payoff profiles (5,0) and (3,3).

The first is better for player A (five instead of three), but worse for player B (zero instead of three).

Neither Pareto-dominates the other. A higher total, or one player’s higher number, does not automatically establish a Pareto improvement.

Coordination games can have multiple equilibria

Now imagine players A and B independently choose R or B as abstract symbols. They both receive two points when their choices match and zero when they differ.

Unlike the Prisoner’s Dilemma, neither player has one dominant choice.

The incentive is to coordinate with the other player’s action rather than choose the same fixed action regardless of the opponent.

The fictional coordination payoff matrix

A’s actionB chooses RB chooses B
A chooses R(2,2)(0,0)
A chooses B(0,0)(2,2)

Every cell is a fictional points outcome. Nothing is being asserted about actual school club selection or classroom peer behaviour.

The matrix is symmetric and rewards matching actions.

Worked example 7: locate the coordination equilibria

At (R,R), either player who switches alone receives zero instead of two, so neither wants to move.

At (B,B), the same reasoning holds. Both profiles are pure-strategy Nash equilibria.

At (R,B), A can switch to B and improve from zero to two; at (B,R), a similar deviation exists. Thus there are two pure equilibria: (R,R) and (B,B).

Why multiple equilibria create a coordination problem

Each matching profile is stable, but the players may need some way to select which one to expect.

Shared conventions or communication can influence coordination, depending on the game’s rules and actual setting.

The equilibrium concept alone does not necessarily specify which of several equilibria will be played.

Best responses can be marked directly in a table

For each fixed B column, identify the largest first payoff, marking A’s best response.

For each fixed A row, identify the largest second payoff, marking B’s best response.

A cell where both players are simultaneously responding optimally is a pure-strategy Nash equilibrium. This method works systematically for small finite payoff matrices.

Worked example 8: inspect the two diagonal cells

At (R,R), A prefers R when B is R, and B prefers R when A is R.

At (B,B), A prefers B when B is B, and B prefers B when A is B.

Thus both diagonal cells satisfy mutual best response. The off-diagonal cells fail because either player can improve by matching the other.

Dominant strategy and Nash equilibrium are not the same idea

A dominant strategy is a player’s best choice regardless of the opponent’s action.

A Nash equilibrium asks whether the chosen strategy for each player is a best response to the other players’ equilibrium strategies.

Some games have Nash equilibria without dominant strategies. The coordination game shows that clearly: neither R nor B dominates, yet two pure equilibria exist.

A game can have no pure-strategy equilibrium

Consider matching pennies. Both players choose H or T. Player A receives +1 if the choices match and −1 if they differ; B receives the negative of A’s payoff.

No pure action pair is stable. If the choices match, B wants to switch. If they differ, A wants to switch.

This illustrates why the broader concept of a mixed strategy is sometimes needed.

The matching-pennies payoff table

A’s choiceB chooses HB chooses T
A chooses H(+1,−1)(−1,+1)
A chooses T(−1,+1)(+1,−1)

This is a fictional zero-sum game because the two payoffs add to zero in every cell.

It is not a real wager or recommendation to gamble.

Worked example 9: why H,H cannot be stable

At (H,H), A receives +1 and B receives −1.

If B alone switches to T, B’s payoff improves to +1. Therefore (H,H) is not a Nash equilibrium.

The same kind of profitable unilateral switch exists in each of the four pure cells, so none is stable.

A mixed strategy assigns probabilities to actions

A mixed strategy is a probability distribution over a player’s available pure actions.

For example, choosing H with probability one-half and T with probability one-half is a mixed strategy.

The probabilities describe deliberate randomisation in the game model. They are not claims that a real student uses a random-number generator or behaves unpredictably.

Expected payoffs let us compare mixed actions

If player B chooses H with probability q and T with probability 1−q, A’s expected payoff from H in matching pennies is

(+1)q+(−1)(1−q)=2q−1.

A’s expected payoff from T is

(−1)q+(+1)(1−q)=1−2q.

Player A is indifferent between H and T when q=1/2.

Worked example 10: solve B’s mixing probability

Set A’s two expected payoffs equal:

2q−1=1−2q.

Then 4q=2, so q=1/2.

When B mixes equally, A cannot improve expected payoff by switching between H and T. By symmetry, B is indifferent when A also mixes equally.

The matching-pennies mixed equilibrium

Each player chooses H with probability 1/2 and T with probability 1/2.

Under this pair of mixed strategies, each player’s expected payoff is zero, and neither can improve it by unilaterally changing their strategy.

This is the unique mixed-strategy Nash equilibrium of this standard two-action zero-sum game.

A probability strategy is not a guaranteed result

Mixing equally does not mean exactly half of every ten plays must be H.

It states a probability rule for choices. Finite realised sequences can contain uneven counts.

Expected payoff describes the distribution over possible outcomes, not the result of each individual play.

Worked example 11: one outcome versus expectation

Suppose both players independently mix equally in matching pennies.

A has probability 1/2 of matching and 1/2 of not matching. Expected payoff is (1/2)(+1)+(1/2)(−1)=0.

A particular round still yields either +1 or −1, not the fractional “zero points” expected value.

Rock–paper–scissors is another zero-sum example

Use abstract moves Rock, Paper and Scissors. A wins one point for winning, loses one for losing and receives zero for a tie. B receives the opposite.

With all three actions available, no pure-strategy pair is a Nash equilibrium under the usual payoff rules.

An evenly randomised strategy is a mixed equilibrium, because no opponent can exploit a preference for one move.

Worked example 12: uniform mixing

Suppose B chooses Rock, Paper and Scissors each with probability one-third.

If A chooses Rock, the expected payoff is (1/3)(tie zero)+(1/3)(lose −1)+(1/3)(win +1)=0.

The same holds for A’s Paper or Scissors. With both players uniformly mixed, neither can improve their expected payoff by choosing a different pure or mixed response.

Predictable mixing may be exploitable

If B selects Rock more often than the other choices, A can sometimes choose a pure move with positive expected payoff.

For example, let B choose Rock with probability 1/2 and Paper and Scissors with probability 1/4 each.

Against that fictional distribution, A’s Paper action wins against Rock with probability 1/2, loses against Scissors with 1/4 and ties Paper with 1/4.

Worked example 13: a best response to biased mixing

Expected payoff of A choosing Paper is (+1)(1/2)+(−1)(1/4)+0(1/4)=1/4.

For A choosing Rock it is 1/4−1/4=0. For Scissors it is 1/4−1/2=−1/4.

Paper gives the highest expected payoff against this stipulated B distribution. This is a hypothetical points calculation, not a strategy recommendation for wagering.

Zero-sum and general-sum games are different

In a zero-sum game, the two players’ payoffs sum to zero for every outcome.

In a general-sum game, both can gain, both can lose or payoffs can move in other patterns.

The coordination and Prisoner’s Dilemma examples above are not zero-sum because their payoff sums differ between cells.

Worked example 14: check whether a game is zero-sum

In matching pennies, (+1,−1) sums to zero and (−1,+1) also sums to zero. Every outcome is zero-sum.

In the coordination game, (2,2) sums to four while (0,0) sums to zero. Therefore it is not zero-sum.

The classification depends on all payoff cells, not the name of the game.

Payoff scales reflect the model’s objective

If each payoff represents invented points, more points may be defined as better for the player.

If payoffs represented a cost, lower numbers might instead be better. The mathematical comparison must reflect the interpretation explicitly.

Calling a table “utility” ordinarily assumes higher values are preferred, but an optimisation problem can be formulated directly with costs.

Worked example 15: change from costs to utility

Suppose two options have costs 3 and 5, with lower cost preferred.

One can define utility as negative cost: −3 and −5. Then maximising utility selects −3, corresponding to cost three.

The choice is unchanged, provided the conversion consistently preserves preferences. Mixing cost-minimising and utility-maximising interpretations would create an apparent contradiction.

A payoff matrix needs complete strategic assumptions

A two-player normal-form game typically assumes the players’ strategy sets and payoff assignments are specified.

It may also assume players make choices simultaneously or without observing each other’s current choice, although normal-form representation can encode other settings through full strategies.

A real interaction may involve incomplete information, repeated negotiation or additional options, requiring a richer model.

A repeated game is not one large isolated choice

Suppose the same two-player interaction occurs in multiple rounds.

Strategies can now depend on past outcomes, provided the rules allow observation and memory. Future consequences can change today’s incentives.

The one-shot dominant action in the original Prisoner’s Dilemma is not by itself a complete description of every repeated interaction under different assumptions.

A finite repeated Prisoner’s Dilemma has a classic logic

In the standard complete-information Prisoner’s Dilemma repeated a fixed known finite number of rounds, the final round retains the one-shot dominant choice D.

Reasoning backwards, future promised cooperation cannot credibly change that final-round incentive in the standard setup. The same argument can repeat to earlier rounds.

This backward induction result depends on the specific game and rationality assumptions. Real behaviour may differ.

An indefinitely repeated game has new incentives

When future rounds have value, a player can compare immediate gain from defection with the cost of losing future cooperative payoffs.

The mathematics depends on how later payoffs are weighted and what strategies are permitted.

A discount factor δ, with 0≤δ<1, is a common way to represent how strongly future per-round utility is valued.

A simple repeated-cooperation calculation

Return to the fictional Prisoner’s Dilemma with cooperation payoff three per round, one-shot temptation payoff five and mutual defection payoff one.

Imagine a grim-trigger rule: both cooperate initially, but a defection causes permanent D play thereafter. Under perfect monitoring and the specified discounted-payoff model, the value of continuing cooperation is

3+3δ+3δ²+⋯=3/(1−δ).

A one-time defection while the other player cooperates yields five now, followed by one per round under the punishment rule.

Worked example 16: compute the temptation value

If a player defects once against cooperation and receives five immediately, then receives one in each future round, the discounted total is

5+δ+δ²+⋯=5+δ/(1−δ).

The formula uses the infinite geometric series for δ between zero and one. It is a model value, not a prediction of actual student behaviour.

Find when cooperation can resist one-shot temptation

Compare the cooperative value 3/(1−δ) with the deviation value 5+δ/(1−δ).

Cooperation is at least as attractive when

3/(1−δ)≥5+δ/(1−δ).

Multiply by positive 1−δ: 3≥5−4δ, so δ≥1/2. Under the stated strategy and monitoring assumptions, sufficient future weight can make cooperation resistant to this unilateral deviation.

Worked example 17: test two discount factors

At δ=0.25, cooperative value is 3/0.75=4, while deviation value is 5+0.25/0.75≈5.333, so the temptation is greater.

At δ=0.75, cooperative value is 3/0.25=12, while deviation value is 5+0.75/0.25=8. Cooperation gives the higher discounted value.

The threshold depends on the specific payoff numbers and continuation strategy, not a universal rule for every repeated game.

A repeated-game equilibrium needs credible continuation

Checking only the initial cooperation incentive is not enough for an arbitrary repeated strategy.

The prescribed behaviour after each possible history must also be rational given the resulting subgame, information and payoff assumptions.

In the standard grim-trigger construction for the Prisoner’s Dilemma, permanent mutual D is a continuation equilibrium because D is dominant in the stage game. That helps support credibility under the stipulated ideal conditions.

Changing one payoff changes the strategic conclusion

Suppose an invented interaction still has choices C and D but the payoff from unilateral defection against C is no longer five.

A dominant-strategy result or repeated-cooperation threshold may change.

The structure follows the complete payoff table. A named game cannot substitute for reading its actual numbers.

Worked example 18: a different temptation value

Keep mutual cooperation at three and mutual defection at one, but change the temptation payoff against C from five to four.

The deviation value is now 4+δ/(1−δ). Comparing with 3/(1−δ) gives 3≥4−3δ, hence δ≥1/3.

A smaller temptation lowers the discount-factor threshold in this simplified grim-trigger model.

Sequential games require a decision tree

In a simultaneous game, each player selects without observing the other’s current action.

In a sequential game, a later mover can observe earlier decisions according to the rules. A game tree represents possible histories and choices.

The order matters because later players may respond differently depending on what occurred earlier.

Construct a fictional sequential game

Player A first chooses Stop or Continue.

If A stops, payoffs are (2,2). If A continues, B chooses Left, giving (0,3), or Right, giving (3,0).

All numbers are fictional utility points. This is an abstract game tree rather than a real classroom command or policy.

Worked example 19: solve B’s later choice

If A chooses Continue, B compares B’s own payoffs.

Left gives B three, while Right gives B zero. B therefore prefers Left.

This is the final decision node. Solving it first is the beginning of backward induction.

Worked example 20: solve A’s first decision

A anticipates that continuing leads B to choose Left, producing A payoff zero.

Stopping produces A payoff two. Therefore A prefers Stop.

A subgame-perfect prediction under the stated information and payoff setup has A stopping, with B’s contingent strategy choosing Left if reached.

Backward induction is more than guessing a likely path

The method solves decisions from the last stage backwards, using each player’s preferences at each relevant choice node.

It does not assume B will select the branch that maximises A’s payoff. B chooses according to B’s own model payoff.

This disciplined reasoning separates a credible future response from a threat or promise that the later player would not actually prefer to execute.

A Nash equilibrium need not be subgame-perfect

In a sequential extensive-form game, some Nash strategy profiles may rely on off-path actions that would not be optimal if those later decisions were actually reached.

Subgame-perfect equilibrium requires equilibrium behaviour in every subgame.

This is an advanced refinement. It matters when discussing credible commitments and backward induction, but it should not be confused with the simpler one-shot normal-form Nash definition.

Repetition and communication may change the game

Communication can provide a coordination signal, alter expectations or introduce commitments.

Whether a message changes payoffs, is enforceable or reveals private information depends on the actual rules.

A statement such as “talking always solves the Prisoner’s Dilemma” is not a theorem without defining what communication permits and how it affects incentives.

Game theory does not prove what people will choose

A Nash equilibrium is a mathematical property of strategies and payoffs.

Real people may have different information, concerns, cultural norms, mistakes or objectives than the model assigns.

One should therefore say “under this payoff model, this profile is an equilibrium” rather than claim that actual Bukit Timah learners will or should select those actions.

Mixed-strategy equilibrium is an expectation condition

A player who randomises between multiple actions in equilibrium must be willing to choose each action used with positive probability.

Typically, those supported actions yield equal expected utility given the other players’ mixed strategies. An unused action cannot yield more.

This explains why solving for mixed-strategy probabilities often involves equating expected payoffs.

Worked example 21: symmetric coordination mixing

In the matching coordination game, B chooses R with probability q and B with probability 1−q.

A’s expected payoff from R is 2q; from B it is 2(1−q). Indifference requires 2q=2(1−q), so q=1/2.

By symmetry, both mixing equally is another Nash equilibrium, alongside the two pure matching equilibria.

Coordination mixing can yield lower expected points

If both players independently choose R or B with probability one-half, they match with probability one-half.

They receive two when matched and zero otherwise, so each player’s expected payoff is 1.

At either pure coordinated equilibrium, each receives two. This illustrates that an equilibrium can be less rewarding than another equilibrium even in a simple coordination game.

Worked example 22: verify the matching probability

With independent fair mixing, the four action pairs RR,RB,BR,BB each have probability one-quarter.

Matching occurs in RR or BB, so probability is 1/4+1/4=1/2.

Expected payoff for either player is (1/2)×2+(1/2)×0=1.

The same payoff sum can hide unequal outcomes

Consider payoff pairs (4,0) and (2,2).

Both have total payoff four, but the distribution differs greatly between players.

Whether one is preferable depends on the objective, fairness criterion or player preferences specified. Game theory can quantify the trade-off but does not automatically dictate a moral standard.

Pareto optimality is not Nash stability

A Pareto-efficient outcome has no alternative feasible outcome making at least one player better off without making someone else worse off.

A Nash equilibrium requires no profitable unilateral deviation given others’ actions.

One is a comparison of outcomes across players; the other is a comparison of unilateral incentives. They can overlap, but neither concept guarantees the other.

Worked example 23: the Prisoner’s Dilemma distinction

At the unique Nash equilibrium (D,D), both receive one.

The feasible outcome (C,C) gives both three, so (D,D) is Pareto-dominated by (C,C).

Thus the Nash equilibrium is not Pareto-efficient. This is not a contradiction because the two definitions ask different questions.

A best response is a conditional decision

If B is certain to choose C in the Prisoner’s Dilemma, A chooses D as the payoff-maximising response.

If B is certain to choose D, A still chooses D in that specific game. In a coordination game, however, A’s best response changes with B’s choice.

Students should calculate a best response for each opponent strategy before searching for equilibrium.

A weakly dominant strategy differs from a strictly dominant one

A strategy is weakly dominant if it does at least as well as every alternative against all opponent choices and strictly better in at least one comparison.

Strict dominance demands strictly better payoff against every opponent choice.

Distinguishing the definitions matters when players are indifferent in some outcomes. Multiple best responses can affect equilibrium counts.

Worked example 24: a weak-dominance example

Imagine A choosing X or Y, while B chooses L or R. Give A payoffs:

X against L: 2; X against R: 2. Y against L: 2; Y against R: 1.

X never gives less than Y and is better against R, but ties against L. Thus X weakly dominates Y, not strictly.

Dominance depends on the player’s own payoff

When deciding A’s strategy, examine only the first payoff in every ordered pair.

When deciding B’s strategy, examine only the second.

A common mistake is comparing the pair sums and calling an action dominant because it raises total welfare. That is a different objective from individual best response.

A payoff matrix should use stable labels

If action R means Red in one row and Right in another part of a problem, ambiguities can arise.

Define every action once and keep the same meaning in each cell. Payoff order must also stay consistent.

A correctly structured table is a tool for reducing confusion, not a substitute for a clear game description.

Worked example 25: interpret a cell carefully

In the coordination game, A chooses R and B chooses B. The payoff cell is (0,0).

If B switches to R while A remains R, the cell becomes (2,2). Both improve in that change.

But the Nash test is performed by examining whether one player can improve at a time, not whether both can benefit from a coordinated simultaneous switch.

Nash equilibrium is a unilateral test, not a coalition test

The ordinary Nash concept holds all other players’ strategies fixed during a deviation.

If two players would benefit only by jointly changing actions, their current profile can still be Nash.

Other equilibrium concepts address coalitions or cooperative agreements, but those require additional definitions and assumptions.

Worked example 26: coordination requires joint change

At (R,R), both receive two, and neither benefits from switching alone.

A joint switch to (B,B) gives them the same payoffs, not more. The original profile remains Nash.

The key point is that the equilibrium test itself focuses on unilateral alternatives regardless of whether other feasible joint changes exist.

A zero-sum game can have a minimax interpretation

In a finite two-player zero-sum game, each player’s objective is opposed to the other’s.

Under the standard minimax theorem assumptions allowing mixed strategies, equilibrium strategies secure a game value against the opponent’s best response.

Matching pennies has game value zero and uniform equilibrium mixing. This result is a property of the mathematical payoff structure, not guidance on financial risk-taking.

Mixed strategies require probability sums of one

If a player assigns probabilities p₁,p₂,… to available actions, every probability must be nonnegative and their sum must be one.

In Rock–Paper–Scissors, uniform mixing assigns 1/3 to each move.

Assigning 1/2 to all three would total 1.5, an invalid probability distribution. An expected-payoff calculation based on such inputs would not describe a valid mixed strategy.

Worked example 27: repair an invalid mix

A pupil proposes P(Rock)=0.5, P(Paper)=0.3, P(Scissors)=0.4.

Their total is 1.2, so the proposed probabilities are invalid.

One possible different valid fictional mix is 0.5,0.3,0.2. Changing the input changes the strategy; it is not merely cosmetic notation.

A mixed strategy does not need every action

A strategy can assign probability zero to some available actions.

For instance, choosing R with probability one and B with probability zero is mathematically the same as the pure strategy R.

A genuinely mixed strategy may place positive probabilities on more than one action. In equilibrium, actions within its support must satisfy the relevant best-response conditions.

Expected utility is a probability-weighted average

If an action produces payoff five with probability p and payoff one with probability 1−p, its expected utility is

5p+1(1−p)=1+4p.

The expected value need not equal any payoff of a single realised outcome.

This connects game theory to the separate Bukit Timah expected value and repeated trials lesson.

Worked example 28: calculate a mixed response payoff

Suppose B chooses C with probability p in the original Prisoner’s Dilemma.

A’s expected payoff from choosing D is 5p+1(1−p)=1+4p.

A’s expected payoff from choosing C is 3p+0(1−p)=3p. Their difference is 1+p>0 for 0≤p≤1, confirming D is still strictly better against every mixed choice by B.

Strict dominance survives mixing by the opponent

If action D gives a strictly higher payoff than C against every pure strategy of the opponent, averaging those differences using nonnegative probabilities preserves strict superiority.

That is why A’s D stays dominant even when B randomises.

This is a general reasoning shortcut. It does not require separately checking every possible probability p when pure-action inequalities already prove strict dominance.

A strategic outcome can be stable and still inefficient

The Prisoner’s Dilemma has a stable (D,D) equilibrium with total two and an attainable cooperative (C,C) outcome with total six.

This illustrates why analysts may study institutions, commitments and repeated interaction as part of the game’s specified rules.

But the mathematics does not guarantee any intervention will work in a real school or society without relevant evidence.

Changing the rules changes the game

Suppose a fictional classroom rule adds a payoff for cooperation, or a binding agreement changes what deviations are possible.

The original matrix no longer fully describes the strategic interaction. Dominance and equilibrium must be recalculated from the revised rules.

A pupil should never reuse an equilibrium from a different table simply because both problems have C and D actions.

Worked example 29: a revised cooperative payoff

Imagine changing A’s payoff at (C,C) from three to six while leaving A’s temptation payoff five against B’s C.

When B cooperates, A now prefers C (six) over D (five). Thus the original claim “D is strictly dominant for A” is false in this revised game.

One numerical change can alter the best-response structure. The result follows from comparing the actual updated payoffs.

Fairness and stability are different modelling choices

One may evaluate an outcome by total points, minimum individual payoff, equality of payoffs or other explicit criteria.

These are different social objectives. Game theory provides tools to reason about incentives, not a universal rule that one fairness measure is correct.

A good educational example names whether it is solving for Nash stability, maximum total payoff or some other objective.

Game theory also has applications beyond two players

Many strategic settings involve several players, each with multiple actions.

The Nash definition remains that no one player can improve through a unilateral deviation. A full payoff table can become very large as the number of combinations grows.

For example, four players each with three actions have 3⁴=81 pure strategy profiles. The multiplication principle determines the count.

Worked example 30: count possible profiles

Suppose three fictional players each choose one of two actions.

There are 2×2×2=8 pure strategy profiles.

With five players and two actions each, there are 2⁵=32 profiles. The number increases exponentially with player count, even before any payoff calculation.

A game tree and an outcome tree are not identical

A game tree describes choices over time and may include information about what each player observes.

A probability tree describes random outcomes and conditional probabilities. Some games contain random moves, but not every branch is a chance event.

A learner must label which nodes belong to decision-makers and which belong to chance if both are present.

Game theory and linear programming answer different questions

Linear programming maximises or minimises a stated objective subject to constraints for a single optimisation problem.

Game theory considers interacting decision-makers whose payoffs depend on each other’s actions.

The Bukit Timah linear programming and corner points guide explores feasible vertices. This guide instead examines strategic best responses and stability.

A local classroom puzzle cannot establish an actual psychological claim

If two fictional players select an action that is not Nash, that does not prove they are irrational.

They may be following different payoffs, misunderstanding instructions, learning the game or prioritising fairness in ways not included in the numerical model.

A sound mathematical conclusion stays within the defined game. Psychological explanations require separate evidence.

Common error 1: swap payoff order

For cell (0,5), A receives zero and B receives five.

If a pupil reverses the players, every best-response calculation may be wrong.

Write “(A,B)” above the table and colour or underline each player’s own payoff when analysing that player’s incentives.

Common error 2: maximise the sum when finding a best response

At (C,C), the combined payoff six exceeds (D,D)’s two.

But Nash stability asks whether one player improves by switching alone, not which total is highest.

The correction is to fix the opponent’s action, compare the chosen player’s payoff, and then test the other player separately.

Common error 3: assume every game has one pure equilibrium

The coordination game has two pure Nash equilibria. Matching pennies has none.

A mixed-strategy equilibrium may exist even when no pure equilibrium exists, under the standard finite-game assumptions.

Do not infer equilibrium count from the number of players or actions without checking payoffs.

Common error 4: assume a dominant action always exists

The coordination game has no action that is best against every possible opponent choice.

A player wants to match the opponent instead.

The definition of a dominant strategy is stricter than that of a best response to one particular opponent strategy.

Common error 5: call expected payoff a guaranteed reward

In matching pennies with uniform mixing, expected payoff is zero, but an individual round gives A +1 or −1.

Expected value describes a probability-weighted average, not the realised outcome of each play.

The correction is to list the possible payoffs alongside their probabilities.

Common error 6: ignore discounting in repeated games

The value of future cooperation depends on how future payoffs are weighted and what continuation strategies are possible.

The threshold δ≥1/2 in our grim-trigger model comes from the exact fictional payoffs and infinite discounting assumptions.

It is not a universal numerical boundary for cooperation in all real or imagined games.

Common error 7: confuse Nash with subgame-perfect equilibrium

A Nash equilibrium tests unilateral deviations in the strategy profile of the whole game.

Subgame-perfect equilibrium additionally requires credible equilibrium behaviour in every relevant subgame of a sequential game.

The distinction matters when a plan specifies actions at decision nodes that might not be reached under the equilibrium path.

Primary readiness: choices and simple tables

Primary learners can list possible pairs of choices, identify outcomes and count how many combinations are available.

They can compare points when an opponent’s choice is fixed, without needing formal equilibrium terminology.

The essential foundation is organised comparison of alternatives and careful reading of the table.

Lower Secondary: probability and best responses

Secondary learners can construct two-action payoff matrices, calculate simple best responses and distinguish an individual payoff from a combined total.

Probability foundations support mixed strategies and expected utility, but there is no need to begin with difficult general theorems.

A tutor should ask pupils to explain which number they are comparing and whose decision can change.

Advanced Mathematics and interdisciplinary context

Game theory belongs to mathematical economics, operations research and strategic decision theory. Detailed Nash equilibria, repeated games and minimax arguments can be advanced enrichment.

The SEAB 2027 G3 Mathematics syllabus directory provides Secondary mathematical prerequisites; the 2027 A-Level syllabus directory provides later pathways.

This article does not claim that Nash equilibrium or repeated-game discounting is universally examinable for Singapore Secondary pupils.

A twenty-minute home diagnostic

Minutes 1–5: Read the Prisoner’s Dilemma payoff table and find A’s best response to B choosing C, then to B choosing D.

Minutes 6–10: Find B’s best responses and verify (D,D) as the unique pure Nash equilibrium. Explain why (C,C) is Pareto-better.

Minutes 11–15: Use the coordination matrix to identify (R,R) and (B,B) as pure equilibria. Explain why there is no dominant matching colour.

Minutes 16–20: In matching pennies, show no pure equilibrium exists, then explain why equal mixing leaves neither player with an expected-payoff advantage.

Seven days of practice and transfer

Day one reads payoff pairs correctly. Day two learns best-response comparisons. Day three identifies dominant strategies.

Day four tests pure Nash equilibria. Day five compares Pareto improvements and total welfare. Day six introduces mixed strategies through expected values.

Day seven constructs a small sequential game tree and works backward from the final decision. The student should justify the outcome in words, not simply name an equilibrium.

Practice questions and checked answers

A. In the stated Prisoner’s Dilemma, A’s best response when B chooses C? Answer: D.

B. A’s best response when B chooses D? Answer: D.

C. Unique pure Nash equilibrium of that game? Answer: (D,D).

D. Total payoff at (C,C)? Answer: six.

E. Pure equilibria of the symmetric 2-point coordination game? Answer: (R,R) and (B,B).

F. Does standard matching pennies have a pure equilibrium? Answer: no.

G. Mixed equilibrium of standard matching pennies? Answer: each player mixes H and T with probabilities 1/2,1/2.

H. In the illustrative discounted grim-trigger calculation, when is cooperation at least as valuable as defecting once? Answer: δ≥1/2 under the stated assumptions.

Frequently asked questions

What is a Nash equilibrium?

A strategy profile in which no player can improve their own payoff by changing only their own strategy while others’ strategies remain fixed.

Is Nash equilibrium the same as the best result for everybody?

No. A Nash equilibrium can be Pareto-inferior to another feasible outcome.

What is a best response?

An action maximising the player’s payoff against specified strategies of the other players.

Is a dominant strategy always available?

No. In coordination games, the best action often depends on what the other player chooses.

Can a game have more than one Nash equilibrium?

Yes. The symmetric coordination game here has two pure equilibria and also a mixed equilibrium.

Can a game have no pure equilibrium?

Yes. Matching pennies has no pure-strategy equilibrium, but does have a mixed-strategy equilibrium.

Does a mixed strategy mean someone always plays each action equally often?

No. It specifies probabilities; finite realised counts can differ from expected frequencies.

Do repeated games always generate cooperation?

No. Results depend on payoffs, information, time horizon, discounting, monitoring and allowed strategies.

Are the payoffs actual Bukit Timah school or tuition rules?

No. Every player, action and payoff is an original fictional mathematical construction.

How can parents recognise understanding?

The student can compare each player’s own payoffs, identify best responses, test unilateral deviations and distinguish strategic stability from welfare or real-world predictions.

Official context and connected Mathematics in Bukit Timah

The SEAB 2027 G3 Mathematics and Additional Mathematics directory provides Singapore school-prerequisite context; the 2027 A-Level directory provides later mathematical pathways. The game tables are entirely fictional and do not describe real pupils, fees or schools.

Read Mathematics in Bukit Timah | Probability and Fair Classroom Games, Expected Value and Repeated Trials, Linear Programming and Optimal Choices, Information Entropy and Efficient Codes and Truth Tables and Implications for separate but connected skills.

The immutable Clementi Secondary 1 Mathematics small-group tutorial reference anchors first-principles teaching, checked examples, diagnosis of reasoning mistakes and independent transfer.

The lasting habit is to define each player’s strategies and payoffs, distinguish self-interest from total welfare, identify every best response, test unilateral deviations and keep equilibrium claims conditional on the stated game rather than invented facts about real people.

More Mathematics in Bukit Timah: this batch’s companion lessons — Taylor Series, Local Approximations and Remainder Bounds · Determinants, Inverse Matrices and Unique Linear Solutions · Confidence Intervals, Standard Errors and Coverage. These four original articles connect local approximation, invertible systems, statistical coverage and strategic decision-making, with worked checks and explicitly fictional examples.