Game theory: when the best move depends on everyone else's
Interactive Nash equilibrium explorer with three financial scenarios: Cournot competition between two firms, a Diamond-Dybvig-style bank run with its tipping point, and the repeated game behind insurance underwriting cycles.
Most models on this site treat uncertainty as nature rolling dice. Game theory handles a different kind of uncertainty: other people, choosing strategically, knowing you are doing the same. The resting point of such a system is the Nash equilibrium, the set of choices where no player gains by changing strategy alone, and it is often collectively worse than what cooperation could achieve.
The three scenarios are financial classics. In the Cournot duopoly two firms compete on quantity: each produces more than its share of the cartel output and both earn less (900 each against 1,012.5 under collusion at the defaults), while a firm that secretly over-produces against a cartel partner earns more still (about 1,139), which is why cartels are unstable even before regulators act. In the bank run, waiting and running are both self-fulfilling equilibria, separated by a tipping fraction of withdrawals (about 0.44 at the defaults); deposit insurance works by removing the bad equilibrium, not by paying claims out of it.
The underwriting cycle treats two insurers' hold-or-cut pricing decisions as a repeated game. When the market share won by cutting alone exceeds the damage of a mutual price war, the single-round game is a prisoner's dilemma and both cut. Rate discipline is sustainable only if the rivalry is likely enough to continue: the continuation probability δ must reach the grim-trigger threshold δ*, 0.625 at the defaults. That is why soft markets are not irrationality but the one-shot equilibrium reasserting itself. Every scenario has sliders, live equilibrium calculations and the payoff logic drawn out.
Cournot best response and Nash equilibrium
q_i* = (m − q_{−i}) / 2, m = a − c → q* = m/3, π* = m²/9
Demand is linear, P = a − Q, where Q is total output; c is each firm's marginal cost, q_i a firm's output and q_{−i} its rival's. Each best-response line gives a firm's profit-maximising output for any rival output, and the lines cross at the Nash point. The defaults a = 120 and c = 30 give q* = 30 and profit π* = 900 per firm.
Frequently asked questions
What is a Nash equilibrium?
A set of strategies, one per player, where no player can improve their payoff by changing strategy while the others hold theirs fixed. It is the standard prediction for the outcome of strategic interaction: stable, but not necessarily efficient.
Why do bank runs happen even when the bank is solvent?
Because the game has two equilibria. If few withdraw, waiting pays more and the bank is fine; if enough withdraw, the bank must sell assets at a loss and late movers get nothing, so running becomes individually rational once others run. A credible deposit guarantee removes the bad equilibrium by making waiting safe regardless of what others do.
What sustains pricing discipline in insurance underwriting cycles?
Repetition. Cutting rates pays in a single round, but in a continuing rivalry the threat of future retaliation can sustain mutual restraint, provided the rivalry is likely enough to continue: δ at or above the grim-trigger threshold s/(s + e), where s is the share gained by cutting alone and e the loss in a mutual price war. Short horizons and high churn tip the market into price wars.
Interactive explorer
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