Supermodular Games

A game in which a player's incentive to raise their action increases with opponents' actions (strategic complements).

Ask the Game theory assistant 1 min read · Updated September 9, 2026

Definition

A smooth game is supermodular if $\partial^2 u_i / \partial s_i \partial s_j \ge 0$ for all $i\ne j$ . For general lattices, $u_i$ has increasing differences in $(s_i,s_{-i})$ and the strategy space is a complete lattice.

The set of Nash equilibria forms a nonempty complete lattice with a largest and smallest element (Topkis, 1979).

Intuition

Actions are complements: if your rival does more, I want to do more too. This contrasts with Cournot (substitutes).

Strategic complements produce monotone best responses — the foundation for comparative statics without uniqueness.

Worked example

Bertrand competition with differentiated goods (prices are complements), bank-run games, and network adoption.

Research joint ventures and arms races exhibit strategic complements.

The math

Topkis-Milgrom-Roberts monotonicity: if best responses are increasing, increasing the opponent's action raises your optimal action.

Milgrom-Roberts (1990): supermodular games have monotone comparative statics and converge under fictitious play.

Where it is used

Industrial organization, macroeconomics, and network effects.

The natural framework for comparative statics in oligopoly and coordination.

More in Game theory

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