Game theory studies decisions whose payoffs depend on the choices of other decision-makers. For a quant, the practical question is whether an execution rule remains sensible when other traders respond to it. A cost curve that stays fixed while you optimise can miss the incentives behind competing quotes, cancellations or order timing.
In a single-agent optimisation problem, the modeller specifies an objective and chooses the best feasible decision. In a game, each player has an objective, and your best choice may depend on their choice. Solving your own problem while holding everyone else's behaviour fixed is only one part of the analysis. Polak's introductory lecture develops this distinction.
Describe the interaction before solving it
A game specifies players, available actions, information, timing and payoffs. A strategy is a plan for choosing actions; in a one-round simultaneous game it can be just one action. In a sequential game, it must also say what to do at the situations the player might face. These distinctions are developed in Yildiz's game-theory notes, §§1.1–1.2.
Payoffs represent what each player values. They can be monetary amounts under an expected-money objective, or utility scores that also reflect risk preferences. The specification matters: the same trading venue can produce different decisions for an inventory-constrained dealer and an investor who must complete an order.
Matching pennies: every fixed choice can be answered
Consider a one-round teaching game. Row and Column simultaneously choose heads (H) or tails (T), without observing the other's choice. Row receives 1 token from Column if the choices match; otherwise Row pays Column 1 token. Both maximise expected tokens. The rules and payoffs are known to both.
Each cell records (Row payoff, Column payoff):
| Row's choice | Column: H | Column: T |
|---|---|---|
| H | ||
| T |
This is a zero-sum game: the payoffs sum to zero in every outcome. That is a property of this example, rather than a requirement of game theory.
No pair of fixed choices is stable against a unilateral change. When they match, Column can switch and turn a loss into a gain. When they differ, Row can switch and do the same. A best response is a strategy that gives the highest expected payoff against the opponent's specified strategy. Here the best-response requirement rules out every pure, or non-randomised, profile. Magazinnik's MIT lecture, slides 30–39, develops this example.
Randomness works only at the right probabilities
A mixed strategy assigns probabilities to the player's pure strategies. Let be Row's probability of H and Column's probability of H. Assume the players randomise independently.
Against Column's mix, Row's expected payoffs from the two fixed actions are
They are equal only at . If , Row prefers H; if , Row prefers T. Similarly, Column is indifferent only when .
Original calculation for the stipulated one-round game. The lines show expected payoffs of fixed actions against Column's mix; they are not realised outcome paths.
With both players independently choosing H with probability one-half, neither can raise expected payoff by changing their own strategy alone. This mutual best-response condition is a Nash equilibrium. Each player's expected payoff is zero; an individual round still pays or .
Arbitrary randomness does not have this property. If both use H with probability , Row's expected payoff is
Holding Column's mix fixed, Row can switch to always H and earn in expectation. The proposed random play therefore fails the equilibrium test. Conversely, when Column uses a fair mix, every Row mix is a best response, including always H. Row must also use a fair mix to make Column best-responding at the proposed equilibrium. One player's indifference alone does not settle the game.
Bring the question back to a trading model
Suppose an execution model treats displayed liquidity as an unchanging supply schedule. That can answer a conditional accounting question: how much would this order cost against this snapshot? A strategic model must additionally specify how liquidity providers and competing traders choose their actions, what they observe and what they value. Matching pennies illustrates the reasoning; its two-action payoff table is not a calibrated market model.
The SelfMadeQuant Order Books lesson gives the concrete starting point: frozen-book walking costs, queue priority, and why cancellations and strategic quoting limit a snapshot's interpretation. The linked page is a public preview; the full lesson requires membership.
An equilibrium is a consistency condition for a specified game. It does not establish that actual traders use its strategies, that learning converges to them, or that the outcome maximises joint welfare. Information, repeated interaction and changing incentives can alter the game itself.
Practice: is the randomness independent?
A modified game transfers 3 tokens on the same matching rule. Before choosing, both players observe the same public signal: H with probability , T with probability . Both copy it. What is Row's expected payoff? Can Column improve by changing only its own rule?
Answer. The choices always match, so Row receives 3 tokens with certainty. Multiplying the two marginal H probabilities would be wrong: they come from the same signal. If the choices instead came from independent draws with those marginals, Row's mean would be tokens. Column can respond to the actual public signal by always choosing its opposite, turning its payoff from into : an improvement of 6 tokens. Randomness shared between players is not the independent mixing used above.
Sources
- Ben Polak, Introduction: Five First Lessons, Yale ECON 159, chapters 1 and 4: strategic situations and the importance of specifying players' payoffs.
- Muhamet Yildiz, Graduate Game Theory, §§1.1–1.2, §1.4.1 and §2.2: game representations, strategies, independent mixing, best responses and equilibrium. The document is dated Spring 2022 and is hosted with MIT's Spring 2024 course.
- Asya Magazinnik, Lecture 3: Mixed Strategy Nash Equilibrium, MIT 17.810/17.811 (2021), slides 30–39: matching pennies, mixed strategies and best-response calculations.