Dynamic Games and Subgame Perfect Equilibrium

Why Are Static Games Not Enough?

In real business and politics, decisions are made sequentially, and later players observe earlier moves. A company decides to enter a market, knowing that the incumbent firm may later initiate a price war. Parties make alternating proposals in negotiations. Countries impose sanctions in response to competitor actions.

In such situations, Nash equilibrium permits "empty threats" — actions that are announced but would not rationally be carried out. The concept of subgame perfect equilibrium (SPNE) eliminates such threats.

Games in Extensive Form and Trees

A dynamic game is depicted by a tree: nodes are decision points, edges are actions, leaves are payoffs. Additionally, information sets (nodes among which the player cannot distinguish) describe incomplete information about the course of the game.

A subgame is a fragment of the tree starting at a single node (singleton information set) and including all its subtrees. A subgame must be "closed": information sets cannot be cut.

Backward Induction

For finite games with perfect information, equilibrium is found using the method of backward induction:

1. Start from the leaves — write out the payoffs
2. At each non-terminal node, the current player chooses the action that maximizes their payoff
3. Replace the subtree with the payoff from this optimal move
4. Move up one level and repeat

Zermelo's theorem (1913): In a finite two-player zero-sum game with perfect information, one of the players has a winning strategy, or both can guarantee a draw. Application to chess: theoretically, one of three situations is predetermined, but computing which one is impossible.

Numerical Example: Market Entry

Monopolist (M) and potential competitor (C). C moves first: Enter (E) or Not Enter (NE). If C has entered, M chooses: Fight (F) or Accommodate (A).

Tree: C → (E → M → {F: (−1, 1), A: (2, 2)}; NE: (0, 4))

Payoffs (C, M). Backward induction: Upon E, M chooses A (2 > 1 for M). C knows this: with E he gets 2; with NE he gets 0. Therefore, C chooses E.

SPNE: C enters, M accommodates. Payoffs: (2, 2).

M's threat "if you enter — I'll start a price war" is incredible: in the subgame "after entry" M prefers to accommodate. This is an incredible threat, which SPNE eliminates.

Subgame Perfect Equilibrium

A strategy profile s* is SPNE if it is a Nash equilibrium in every subgame, including the entire game.

Difference from Nash equilibrium: NE allows threats that are never carried out. SPNE requires that each decision be optimal both on and off the equilibrium path.

Ultimatum game: Player 1 proposes how to split 100 rubles (x, 100−x). Player 2 accepts or rejects. If rejected — both receive 0.

Backward induction: Player 2 accepts any x > 0 (better than 0). Player 1 will offer x = 1 (the minimum Player 2 will accept). SPNE: (1, 99) and acceptance of any x ≥ 1.

Experiments: real people offer 40–50% and reject offers below 20–30%. "Unfair" offers are rejected contrary to SPNE. This is a "behavioral" anomaly — a violation of pure material rationality due to a sense of fairness.

Repeated Games and Cooperation

In the infinitely repeated prisoner’s dilemma, cooperation can be sustained by the "grim trigger" strategy: cooperate until the first betrayal, then betray forever. Condition for cooperation: δ ≥ (b−a)/(b−d), where a — mutual cooperation, b — payoff of the betrayer, d — mutual betrayal.

Folk theorem: In an infinitely repeated game with sufficiently patient players (δ → 1), any feasible and individually rational payoff vector can be sustained as SPNE. This explains "tacit collusion" — firms do not explicitly agree, but maintain high prices through the threat of price wars.

Computer chess implements backward induction via the minimax algorithm with alpha-beta pruning — this is SPNE explicitly on the game tree. AlphaZero replaced explicit tree search with neural network position evaluation, but the principle of evaluating "from the end" — the expected outcome under best play by both sides — was retained via Monte Carlo tree search.

SPNE in Corporate Strategy and Negotiations

Subgame perfect equilibrium is applied in a multitude of real situations. In corporate negotiations, the Rubinstein model with alternating offers predicts: under equal discounting the parties split the sum exactly in half, while the less patient party receives a smaller share. This explains why urgent need for a deal weakens bargaining position. In pricing strategy, large firms sometimes set intentionally low prices prior to the appearance of a competitor — so-called "limit pricing." Backward induction shows when this strategy is rational: only if a price war is truly preferable to duopoly for the dominant firm. If the competitor’s fixed costs are high, the incredible threat of war may suffice. SPNE theory is also used in antitrust regulation: regulators model the merger tree to predict the unified company’s behavior and assess harm to competition without actual merger experience.

Subgame Perfect Equilibrium in Corporate Strategy

SPNE is used in structuring corporate deals and competitive strategies. Merger negotiations with alternating offers are modeled as the Rubinstein game: the more patient side (with a lower discount rate) receives a larger share of the created value. This explains why companies under financial pressure are sold at a discount — their "patience" is lower. In market entry strategy, the incumbent's threat of a price war prevents entry only if it is credible in the context of SPNE. If a price war would cost the incumbent more than admitting a new competitor, the threat is not SPNE and the entrant should enter. This underlies "signaling" strategies: incumbents invest in excess capacity to make the threat of war credible. Investments in reputation, creating a "bridge" via long-term contracts, and public commitments — all of these are mechanisms for turning incredible threats into credible ones, changing the SPNE in favor of the incumbent.

Assignment: Build a tree for the "ultimatum" game with three possible offers: (10, 90), (50, 50), (90, 10). Find all Nash equilibria and all SPNE. Explain why there are more NE than SPNE.