Backward Induction
Backward induction is a method of reasoning used in game theory and dynamic optimisation to solve sequential problems. It proceeds by analysing the final stage of a decision problem first and then working backward to the initial stage, determining the optimal action at each point given the actions that will follow.
Method
The method begins with the last decision to be made and determines the optimal choice at that point, given the conditions that prevail. With the outcome of the final stage determined, the method moves to the preceding stage and determines the optimal choice there, and so on, until the initial decision is reached.
Backward induction is used to solve finite extensive-form games, in which players move in sequence and the structure of the game is known. The solution concept associated with the method is subgame perfect equilibrium, which requires that the strategy be optimal at every point in the game, not only at the beginning.
Applications in Game Theory
Backward induction is applied to bargaining games, to entry deterrence, and to repeated games with a known endpoint. In the centipede game, for example, backward induction predicts that players will terminate the game immediately, even though cooperation would yield higher payoffs, a result that has been debated because it conflicts with observed behaviour.
The method is also used in the analysis of strategic interactions in which timing matters, including sequential auctions, pricing decisions, and investment games. It clarifies the conditions under which commitments are credible and the strategic value of moving first or last.
Applications in Economics
Backward induction is used in dynamic programming, the technique for solving optimisation problems that unfold over time. The method determines the value function at each stage and the optimal policy, working backward from the final period.
The approach is applied to consumption and saving decisions over the life cycle, to investment under uncertainty, and to the design of contracts and mechanisms. It provides the foundation for the analysis of dynamic problems in which decisions at one time affect the options available later.
Limitations
The limitations of backward induction include its reliance on common knowledge of rationality and on the assumption that players will choose optimally at every point. Experimental evidence shows that individuals often deviate from the predictions of backward induction, particularly in games with long horizons, and that behaviour is influenced by fairness and by other considerations.
The method also has limited applicability in games with infinite horizons, where there is no final stage from which to begin. In such games, alternative solution concepts, including stationary strategies and Markov perfect equilibrium, are used. The debate about the descriptive accuracy of backward induction continues, and it informs the analysis of behaviour in strategic settings.
Subgame Perfect Equilibrium
Subgame perfect equilibrium is the solution concept obtained by backward induction in extensive-form games. A strategy profile is subgame perfect if it constitutes a Nash equilibrium in every subgame of the original game. The concept refines Nash equilibrium by eliminating strategies that rely on non-credible threats.
The refinement matters because Nash equilibrium in an extensive-form game may involve threats that would not be carried out if the threatened contingency arose. Backward induction eliminates such threats by requiring optimality at every point in the game.
Credibility and Commitment
Backward induction clarifies the conditions under which threats and promises are credible. A threat is credible only if it would be optimal to carry it out when the contingency arises. Where carrying out the threat would be costly, the threat lacks credibility, and rational opponents will not be deterred.
The analysis of credibility informs the design of commitments, including contracts, reputation, and delegation. By making a costly action automatic or by tying one's hands, a party can make a threat credible and improve its strategic position.
Applications in Industrial Organisation
Backward induction is applied to the analysis of entry deterrence, pricing, and investment. In entry deterrence, an incumbent may invest in capacity or in a reputation for aggressive behaviour to deter entry, and the credibility of the deterrence depends on whether the incumbent would actually fight entry.
The analysis informs the assessment of strategic behaviour and of the effects of market structure on outcomes. It clarifies the conditions under which firms can sustain market power and the circumstances in which entry occurs.
Behavioral Evidence
Experimental evidence on backward induction is mixed. In games with short horizons and clear structure, players often approximate the predictions of backward induction. In games with long horizons, players deviate, and the outcome is often more cooperative than the theory predicts.
The deviations have motivated models that incorporate bounded rationality, fairness, and learning. These models provide a more accurate account of behaviour in strategic settings, and they inform the analysis of economic institutions and of market outcomes.