Real-World Mathematics: Game Theory, Payoffs, Nash Equilibrium, Mixed Strategies and Auctions

Application of Mathematics in Real-World Usage · Guide 44 · BTT Mathematics Hub

Game theory studies decisions whose outcomes depend on what more than one decision-maker chooses. The Mathematics is not about guessing personalities. It starts by declaring players, strategies, information and payoffs, then asks which choices are best responses to other choices.

This guide uses fictional payoff tables, bids and valuations for Mathematics teaching. It is not business, investment, procurement, gambling or auction advice. Real strategic settings involve law, institutions, incomplete information, repeated interaction, behavioural effects and objectives that a small payoff matrix cannot capture.

A payoff matrix turns strategic interaction into numbers

Consider two players, Row and Column, each choosing A or B. Suppose Row’s payoffs are:4 at(A,A),0 at(A,B),0 at(B,A),2 at(B,B). Column has the same coordination payoffs.

The table says how each joint choice is scored. It does not itself say what either player will choose.

A best response maximises payoff against a fixed opponent choice

If Column chooses A in the coordination example, Row compares payoff4 from A with0 from B, so A is Row’s best response.

If Column chooses B, Row compares0 from A with2 from B, so B is the best response.

A Nash equilibrium is a profile of mutual best responses

A strategy profile is a Nash equilibrium when no single player can improve their payoff by changing strategy alone while the others keep theirs fixed.

In the coordination matrix above, both(A,A) and(B,B) are pure-strategy Nash equilibria.

Dominance compares one strategy across every opponent choice

A strategy strictly dominates another if it yields a higher payoff against every possible opponent strategy.

For a fictional Row payoff table where U gives3 against L and2 against R while D gives1 against L and0 against R, U strictly dominates D.

Eliminating a strictly dominated strategy reduces the game

If a strategy is strictly dominated, it cannot be a best response to any opponent strategy in the complete-information one-shot model.

Removing it can make a larger payoff table easier to analyse, but weak dominance requires more care because ties can change conclusions.

Zero-sum games make one player’s gain the other’s loss

In a zero-sum representation, Column’s payoff is the negative of Row’s payoff. One payoff matrix therefore contains the whole numerical game.

This special structure supports minimax reasoning, but many real interactions are not zero-sum because both sides can gain or lose together.

Mixed strategies are probability distributions over actions

A player using probability p for U and1−p for D is not choosing a fractional action. They are randomising between whole actions according to a probability rule.

Expected payoff is the probability-weighted average across possible outcomes.

Indifference equations reveal mixed equilibrium probabilities

Take a zero-sum Row payoff matrix [[4,0],[0,2]]. Let Column play L with probability q and R with1−q.

Row’s expected payoff from U is4q. From D it is2(1−q). For Row to mix, set them equal:4q=2(1−q), giving q=1/3.

By the symmetric calculation, Row plays U with probability p=1/3. The game value is4/3.

Expected payoff is linear in probabilities

If a strategy produces payoff10 with probability0.3 and payoff2 with probability0.7, expected payoff is0.3×10+0.7×2=4.4.

Expectation is not a guarantee that4.4 will occur in any single play; it is a long-run probability-weighted mean.

Sequential games add a time order

In an extensive-form game, later choices can depend on earlier observed moves. Backward induction begins at the final decision nodes and works backward toward the start.

If a second mover would choose X for payoff5 rather than Y for payoff2 after a particular branch, the first mover can use that predicted best response when comparing the earlier branches.

Subgame-perfect equilibrium rules out unsupported threats in the model

A subgame-perfect Nash equilibrium requires equilibrium play in every subgame, including branches that are not reached on the equilibrium path.

This sharpens sequential-game analysis because a claimed future action must still be optimal if that decision point is actually reached.

Auctions are games because a bid’s outcome depends on other bids

In a single-item first-price sealed-bid teaching model, the highest bidder wins and pays their own bid.

If a bidder values the item at100, bids70 and wins, their simplified payoff is100−70=30. If the model assigns a60% probability of winning at that bid, expected payoff is0.6×30=18 before any other costs.

A higher first-price bid trades payoff margin for win probability

Bidding90 instead of70 leaves only10 payoff if successful, but it may change the probability of winning. The optimum therefore depends on beliefs about other bids and the auction model.

A bid cannot be evaluated from its amount alone; expected payoff combines both outcome payoff and outcome probability.

Second-price auctions use a different payment rule

In the standard single-item private-value second-price model, the highest bidder wins but pays the second-highest bid. Truthful bidding is a weakly dominant strategy under the textbook assumptions.

If a bidder’s value is100, they win and the second-highest bid is82, simplified payoff is18 regardless of how far their winning bid lies above82, provided the bid remains high enough to win.

Bayesian games model uncertainty about other players’ types

A type can encode private information such as a valuation. A strategy then maps each possible type to an action.

Bayesian Nash equilibrium requires each type’s strategy to maximise expected payoff given beliefs about other players’ types and strategies.

Repeated interaction can make future payoff matter today

A payoff received one period ahead is often represented with discount factor δ between0 and1. A stream1,δ,δ²,… sums to1/(1−δ) when |δ|<1.

At δ=0.9, the infinite geometric sum is10. Repeated-game incentives depend on the specified continuation strategies, not on the discount factor alone.

Pareto comparison and Nash equilibrium answer different questions

Nash equilibrium asks whether unilateral deviation pays. Pareto comparison asks whether one outcome can improve at least one player without hurting another.

An equilibrium need not maximise the sum of payoffs, and a high-total-payoff outcome need not be stable against unilateral deviation.

A complete game-theory calculation states information and timing

State players, available actions, payoffs, move order, what each player knows, whether randomisation is allowed, and whether interaction is one-shot or repeated.

The return path is strategic setting → game representation → best responses → equilibrium or optimisation calculation → interpretation under the model’s assumptions.

Practice: twenty game-theory Mathematics questions

1. In the coordination game, if Column chooses A, compare Row payoffs4 from A and0 from B. What is Row’s best response?
2. If Column chooses B, compare Row payoffs0 from A and2 from B.
3. Name the two pure Nash equilibria in the symmetric coordination game.
4. If U gives3 vs L and2 vs R while D gives1 vs L and0 vs R, which Row strategy strictly dominates?
5. In a zero-sum game, if Row payoff is+5, what is Column payoff?
6. For matrix [[4,0],[0,2]], set4q=2(1−q). Find q.
7. Find Row’s symmetric mixing probability p.
8. Find game value4q.
9. Payoff10 with probability.3 and2 with probability.7: find expectation.
10. Why is an expected payoff of4.4 not a guaranteed single-play result?
11. In a final node, X pays5 andY pays2. Which action does backward induction select there?
12. Value100,bid70,win: find first-price payoff.
13. If win probability is.6, find expected payoff from question12.
14. Value100, second-price payment82: find simplified winner payoff.
15. At discount factor.9, find1+.9+.9²+….
16. What equation defines a Nash equilibrium conceptually?
17. Why does a mixed strategy use probabilities rather than fractional actions?
18. Why can a higher first-price bid lower payoff conditional on winning?
19. Why does a Bayesian game need beliefs about hidden types?
20. Why can a Nash equilibrium fail to maximise total payoff?

Worked answers

1. A.
2. B.
3. (A,A) and(B,B).
4. U.
5. −5.
6. 1/3.
7. 1/3.
8. 4/3≈1.333.
9. 4.4.
10. Because expectation is a probability-weighted average across repeated or uncertain outcomes.
11. X.
12. 30.
13. 18.
14. 18.
15. 10.
16. Each player’s strategy must be a best response to the others, so no player gains by deviating alone.
17. Because randomisation selects whole actions according to a probability distribution.
18. Because payoff margin value−bid becomes smaller as the bid rises.
19. Because expected best responses depend on uncertainty about other players’ private information.
20. Because equilibrium is about unilateral incentives, not maximising the sum of all players’ payoffs.

Sources and connected applications

For strategic-form games, Nash equilibrium, mixed strategies and sequential games, see MIT OpenCourseWare: Economic Applications of Game Theory and MIT Game Theory lecture slides. MIT’s course also treats auctions as strategic games. All payoffs, bids and valuations here are fictional teaching examples.

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