Summary
Game theory is the mathematical study of strategic decision-making among rational actors. It is relevant to game designers in two ways: as a tool for predicting and designing player behaviour in competitive or social situations, and as a source of structural patterns (payoff matrices, dominant strategies, equilibria) that directly map onto game mechanics. Chapter 21 of Hiwiller’s textbook also covers its limits — real players are not rational actors — which is equally important.
(Hiwiller, Players Making Decisions, see source-players-making-decisions)
Key ideas
Payoff matrices
A payoff matrix represents a game in “normal form”: rows are one player’s strategies, columns are another’s, and each cell contains the payoffs for both players as a tuple (Player 1 payoff, Player 2 payoff).
Example — the classic Prisoner’s Dilemma:
| B Rats | B Stays Quiet | |
|---|---|---|
| A Rats | (3, 3) | (0, 5) |
| A Stays Quiet | (5, 0) | (2, 2) |
(Numbers are years in prison — lower is better)
Strict dominance
A strategy strictly dominates another if it produces a better outcome for the player regardless of what any other player does.
In the Prisoner’s Dilemma above, “Rat” strictly dominates “Stay Quiet” for both players: whether B rats or stays quiet, A always gets a shorter sentence by ratting. Since the game is symmetric, B faces the same incentive. Both players rat — getting (3, 3) — even though cooperation would produce (2, 2), which is better for both.
Iterated Elimination of Dominated Strategies (IEDS): When a matrix has no immediately obvious solution, iteratively remove strictly dominated strategies until equilibria are revealed. Each elimination changes which other strategies can be dominated in the next round.
Nash equilibrium
A Nash equilibrium is a combination of strategies where no single player can improve their payoff by changing their strategy while all other players hold theirs fixed. It is the “stable” outcome of a game — the result rational players converge on when each optimises individually.
In the Prisoner’s Dilemma, (Rat, Rat) → (3, 3) is the Nash equilibrium. No player can unilaterally do better by switching — if A switches to Quiet while B Rats, A goes from 3 years to 5 years.
Pareto optimality
A Pareto optimal outcome is one where no player can be made better off without making another player worse off. In the Prisoner’s Dilemma, (Stay Quiet, Stay Quiet) → (2, 2) is Pareto optimal — it is the best collective outcome. But it is not the Nash equilibrium. This is the dilemma: rational self-interest produces a collectively worse outcome.
Zero-sum games
In a zero-sum game, one player’s gain is exactly another player’s loss — resources transfer, the total does not change. Rock, Paper, Scissors is zero-sum (winning takes the round from the opponent). Many head-to-head competitive games are zero-sum or near-zero-sum.
Minimax: In all two-player zero-sum games, the minimax strategy is optimal: minimise the maximum loss (or equivalently, maximise the minimum gain). This is the rational solution for zero-sum games. It does not apply to non-zero-sum games.
The Stag Hunt (coordination games)
The Stag Hunt is a coordination game where both players benefit from cooperating but each risks being worse off if they cooperate while the other defects. Unlike the Prisoner’s Dilemma, the Stag Hunt has a Pareto optimal outcome that is also a Nash equilibrium — cooperation is stable if both players trust each other. Games involving guilds, alliances, or team objectives often have a Stag Hunt structure: everyone is better cooperating, but individual safety favours defection if trust is uncertain.
Mixed strategies
Some games have no pure Nash equilibrium — any pure strategy can be exploited. Rock, Paper, Scissors is the textbook example: if either player commits to a pure strategy (always play Rock), the opponent can exploit it. The Nash equilibrium is a mixed strategy: randomise between options with specific probabilities.
Mixed strategies are relevant to game design in competitive games: if any pure strategy dominates, skilled players converge on it and the game loses depth. Balancing a game so that multiple strategies are viable is equivalent to engineering a mixed-strategy Nash equilibrium where players must diversify rather than exploit a dominant option.
Sequential games
Sequential (extensive form) games involve players taking turns, with each player knowing what came before. They are represented as game trees rather than matrices. The solution method is backward induction — solve from the end of the tree backwards to determine rational play at each node.
Sequential games raise the additional issues of commitment and promises: a player may benefit from credibly committing to a future action (e.g. “I will always retaliate”), but credibility requires that the commitment be genuinely costly to reverse.
Iterated games
When a game is played repeatedly between the same players, cooperation can emerge even in Prisoner’s Dilemma situations — because players can punish defection in future rounds. The most robust strategy in iterated Prisoner’s Dilemma experiments is Tit-for-Tat: cooperate on the first round, then mirror whatever the opponent did last round. It is simple, forgiving, and resistant to exploitation. Multiplayer online games with persistent reputations and ongoing interactions often produce Tit-for-Tat dynamics emergently.
Limits: real players are not rational actors
Chapter 21 of Hiwiller’s textbook notes critical failures of the rational actor assumption:
- The Dollar Auction: Players routinely bid more than a dollar for a dollar, escalating past rational stopping points due to sunk cost reasoning.
- The “Guess Two-Thirds” Game: Rational players should guess 0 (if everyone reasons iteratively), but real players typically guess 22–33, because they assume opponents reason only 1–2 levels deep.
- Second-Price Auctions: While optimal bidding is theoretically simple (bid your true value), real players routinely deviate in ways that reflect psychological biases rather than rational calculation.
Implication: Game theory predicts what rational players do. Real players use heuristics, reason only a few levels deep, and are subject to cognitive biases (see meaningful-decisions). Game theory is a useful design tool for structural analysis, but playtesting against real players is essential — the equilibria your math predicts may not match actual play.
Applying game theory to design
Identifying dominant strategies: If playtesting reveals that all skilled players converge on one strategy, your game likely has a dominant strategy in game theory terms. This is a balance problem — the dominant strategy eliminates meaningful decision-making.
Designing for mixed-strategy equilibria: Viable alternative strategies are the goal in competitive games. Each approach should have specific situations where it outperforms others — i.e. no single strategy should dominate.
Prisoner’s Dilemma patterns in multiplayer: Competitive games where defection (playing selfishly) benefits individuals but hurts the whole often have Prisoner’s Dilemma structures. Common examples: betrayal mechanics, resource hoarding, kingmaking. Designers can engineer away from the equilibrium by adding punishment mechanisms, reputation systems, or repeated play dynamics.
Stag Hunt in cooperative games: Cooperative game modes where players benefit from coordination but can individually “free-ride” have Stag Hunt structure. Building trust mechanics, communication tools, or shared-failure conditions can strengthen the cooperative equilibrium.
Evidence
“A game where the equilibrium is also Pareto optimal is called deadlock. In deadlock, both players end on an outcome that is mutually beneficial.” — Hiwiller, Ch. 19, see source-players-making-decisions
On dominant strategies: if a player would always prefer one strategy regardless of what the opponent does, that strategy “dominates” all alternatives and, if rational actors are assumed, the game is effectively solved. “Any option that is dominated cannot be equilibrium. If an option is dominated, it means that something else is always better, so why would the players ever pick it?” — Hiwiller, Ch. 19.
Implications
- A game with a dominant strategy is “solved” for skilled players — they will converge on it, collapsing the decision space.
- Game theory provides a vocabulary for describing balance problems (dominant strategies, dominated options, equilibria) that is more precise than informal descriptions.
- Emergence and player creativity often exploit game states outside the designer’s anticipated equilibria — this is not always a problem.
- The rational actor model is a useful approximation, not a description of real players. Pair game theory analysis with playtesting.
Open questions
- Most game theory assumes complete information. Many games involve hidden information (fog of war, private hands, hidden roles). How do the equilibria change under incomplete information?
- If real players reason only 1–2 levels deep in strategic situations, does designing for “deep” game theory equilibria produce games only theorists can appreciate?
Related
- meaningful-decisions — Dominant strategies are a form of obvious decision; trade-offs correspond to mixed-strategy equilibria
- game-balance — Balance problems often have game theory descriptions; dominant strategies violate competitive balance
- randomness-in-games — Mixed strategies in game theory are formalised randomness; variance compresses skill gaps
- mda-framework — Emergent dynamics arise from player strategic interaction; game theory describes the structure of that interaction
- reward-systems — Iterated Prisoner’s Dilemma dynamics emerge in games with persistent rewards and reputation
- internal-economy — Economy creates resource competition structures that game theory can analyse
- source-players-making-decisions