Forward Induction Research Articles

On the Quadratic Convergence of Newton’s Method for Mean Field Games with Non-separable Hamiltonian

AbstractWe analyze asymptotic convergence properties of Newton’s method for a class of evolutive Mean Field Games systems with non-separable Hamiltonian arising in mean field type models with congestion. We prove the well posedness of the Mean Field Game system with non-separable Hamiltonian and of the linear system giving the Newton iterations. Then, by forward induction and assuming that the initial guess is sufficiently close to the solution of problem, we show a quadratic rate of convergence for the approximation of the Mean Field Game system by Newton’s method. We also consider the case of a nonlocal coupling, but with separable Hamiltonian, and we show a similar rate of convergence.

Sophisticated banking contracts and fragility when withdrawal information is public

I study whether self‐fulfilling bank runs can occur when banks use sophisticated contracts and withdrawal decisions are public information. In a finite‐agent version of Diamond and Dybvig (1983) with correlated types, I first present an example in which a bank run perfect Bayesian equilibrium exists. However, its existence relies on off‐path beliefs that are unreasonable in terms of forward induction. To discipline beliefs, I use forward induction equilibrium (Cho (1987)) as the solution concept. I show that, whenever the allocation rule is strictly incentive compatible, the truth‐telling strategy is the unique forward induction equilibrium in the withdrawal game, and no bank run occurs. Therefore, with forward induction, sophisticated contracts can prevent bank runs when there is public information about withdrawal decisions.

Forward induction and market entry with an endogenous outside option

Forward induction and market entry with an endogenous outside option

Equilibrium selection through forward induction in cheap talk games

This paper provides a refinement that uniquely selects the ex-ante Pareto dominant equilibrium in a cheap talk game, provided one exists. The refinement works by embedding any cheap talk game into a class of two-stage games where: in stage 1 sender and receiver can, at a cost, alter their preferences; and in stage 2 the cheap talk game is played. For such games, we show that a forward induction logic can be invoked to select the ex-ante Pareto-dominant equilibrium in the second stage. Cheap talk games with exogenously fixed preferences are then treated as limiting cases of this larger class of games.

Forward Induction and Dynamic Optimization Under Uncertainty

Forward Induction and Dynamic Optimization Under Uncertainty

Observability, dominance, and induction in learning models

Learning models do not in general imply that weakly dominated strategies are irrelevant or justify the related concept of “forward induction,” because rational agents may use dominated strategies as experiments to learn how opponents play, and may not have enough data to rule out a strategy that opponents never use. Learning models also do not support the idea that the selected equilibria should only depend on a game's reduced normal form. However, playing the extensive form of a game is equivalent to playing the normal form augmented with the appropriate terminal node partitions so that two games are information equivalent, i.e., the players receive the same feedback about others' strategies.

Learning and total evidence with imprecise probabilities

In dynamic learning, a rational agent must revise their credence about a question of interest in accordance with the total evidence available between the earlier and later times. We discuss situations in which an observable event F that is sufficient for the total evidence can be identified, yet its probabilistic modeling cannot be performed in a precise manner. The agent may employ imprecise (IP) models of reasoning to account for the identified sufficient event, and perform change of credence or sequential decisions accordingly. Our proposal is illustrated with four case studies: the classic Monty Hall problem, statistical inference with non-ignorable missing data, frequentist hypothesis testing, and the use of forward induction in a two-person sequential game.

Equilibrium Selection Through Forward Induction in Cheap Talk Games

Equilibrium Selection Through Forward Induction in Cheap Talk Games

The power of deterministic option-implied trees in pricing European options

ABSTRACT The aims of the current article are threefold. First, to investigate the power of deterministic option-implied trees, constructed either by forward or by backward induction, in pricing European options, in order to assess the proper representation of the smile. Second, to investigate and contrast the power of deterministic option-implied trees during tranquil and volatile market conditions. Last, to assess the correctness of the representation of the smile in different parts of the risk-neutral distribution. Three main results are obtained. First, the pricing performance of the Enhanced Derman and Kani model (EDK), based on forward induction, is superior to that of the Rubinstein model, based on backward induction. Second, the EDK model produces better results (smaller errors) on the left tail of the distribution, i.e. it is better in pricing out-of-the-money put options. Third, it performs better in turmoil periods where correct pricing a challenge, and accuracy is of greater importance than in tranquil periods. Diebold and Mariano test of equal predictive accuracy confirms the superiority of the EDK model in both sub-periods.

Delegation Using Forward Induction

This paper explores how delegation can be used as a signal to sustain cooperation. I consider a static principal–agent model with two tasks, one resembling a coordination game. If there is asymmetric information about the agent’s type, the principal with high private belief can delegate the first task as a signal. This is also supported by the forward induction argument. However, in the laboratory setting, this equilibrium is chosen only sometimes. When the subjects have information about past sessions, there is a significant increase in the use of delegation. This finding sheds light on equilibrium selection in Bayesian games.

Strategic stability of equilibria in multi-sender signaling games

The concept of unprejudiced beliefs equilibrium is simple: out-of-equilibrium beliefs should be consistent with the principle that multiple deviations are infinitely less likely than single deviations. Our questions are: does there always exist such an equilibrium and what can be done if there are multiple such equilibria?To select a unique equilibrium, this notion is usually coupled with the intuitive criterion. The simultaneous usage of these concepts is ad hoc, unjustified, and again might eliminate all the equilibria.We show that coupling these notions is legitimate, as both are implied by strategic stability (Kohlberg and Mertens (1986)), hence a desired equilibrium always exists. The intuitive criterion is trivially implied by stability. We show that in generic multi-sender signaling games stable outcomes can be supported with unprejudiced beliefs. It follows by forward induction that stable sets contain an equilibrium which is unprejudiced and intuitive at the same time.

Information Disclosure and Financial Fragility

I study how banks and other financial intermediaries can use information disclosure to prevent self-fulfilling bank runs. I begin with a finite-agent version of Diamond and Dybvig (1983) with correlated liquidity shocks and sequential service. I allow the intermediary to inform each investor about the withdrawal decisions of previous investors. Adding information disclosure creates a withdrawal game with sequential signaling, and I argue using examples that it is natural to introduce an equilibrium concept placing restrictions on agents’ off equilibrium beliefs. I use the concept of forward induction equilibrium (Cho, 1987) that generalizes the “intuitive” criterion. I provide conditions under which the induced withdrawal game has a unique forward induction equilibrium and no bank run occurs. In other words, disclosing withdrawal information can promote financial stability.

Self-enforcing Agreements and Forward Induction Reasoning

Abstract In dynamic games, players may observe a deviation from a pre-play, possibly incomplete, non-binding agreement before the game is over. The attempt to rationalize the deviation may lead players to revise their beliefs about the deviator’s behaviour in the continuation of the game. This instance of forward induction reasoning is based on interactive beliefs about not just rationality, but also the compliance with the agreement itself. I study the effects of such rationalization on the self-enforceability of the agreement. Accordingly, outcomes of the game are deemed implementable by some agreement or not. Conclusions depart substantially from what the traditional equilibrium refinements suggest. A non-subgame perfect equilibrium outcome may be induced by a self-enforcing agreement, while a subgame perfect equilibrium outcome may not. The incompleteness of the agreement can be crucial to implement an outcome.

Communication Via Delay in a Coordination Game

This paper studies an option of delay in coordination games, allowing players to wait and then choose between the risk-dominant and payoff-dominant actions. The delay option enables forward induction whereby a player’s waiting signals an intention to choose the payoff-dominant action later. If players have e-social preferences (they help other players if at no cost to themselves), iterated weak dominance yields an outcome where everyone waits and then chooses the payoff-dominant action if everyone else waited, so that efficient coordination results. Experimental evidence confirms that waiting can signal future play of the payoff-dominant action.

On non-monotonic strategic reasoning

Strong-Δ-Rationalizability introduces first-order belief restrictions in the analysis of forward induction reasoning. Without actual restrictions, it coincides with Strong Rationalizability (Battigalli, 2003; Battigalli and Siniscalchi, 2003). These solution concepts are based on the notion of strong belief (Battigalli and Siniscalchi, 2002). The non-monotonicity of strong belief implies that the predictions of Strong-Δ-Rationalizability can be inconsistent with Strong Rationalizability. I show that Strong-Δ-Rationalizability refines Strong Rationalizability in terms of outcomes when the restrictions correspond to belief in a distribution over terminal nodes. Moreover, under such restrictions, the epistemic priority between rationality and belief restrictions is irrelevant for the predicted outcomes.

Conditional Pure Strategy Dominance

Given a dynamic game with ordinal preferences, we deem a strategy sequentially rational if there exist a von Neumann-Morgenstern utility function that agrees with the assumed ordinal preferences and a conditional probability system with respect to which the strategy is a maximizer. We prove that this notion of sequential rationality is characterized by a notion of dominance, called Conditional B-Dominance, that extends Pure Strategy Dominance of Borgers (1993) to dynamic games represented in their extensive form. Additionally, we introduce an iterative procedure based on Conditional B-Dominance with a forward induction reasoning flavour, called Iterative Forward B-Dominance, that we prove satisfies nonemptiness. Finally, we show that Iterative Forward B-Dominance selects the unique backward induction outcome in generic dynamic games with perfect information.

On the strategic value of \u2018shooting yourself in the foot\u2019: an experimental study of burning money

This paper uses a novel experimental design to investigate the strategic value of making a public payoff sacrifice (“burning money”) in a setting modelled after the battle-of-the-sexes game. Unlike prior studies, we find that subjects choose to burn money in a significant portion of decision trials and that burning makes the first movers more likely to achieve their preferred stage two equilibrium outcome. No such effect is observed in a control treatment where money burning is triggered unintentionally. This suggests that payoff sacrifices are important as deliberate signals of one’s further intentions, and can be a source of strategic advantage different from that stemming from being the first party to move. These results are supported by an analysis of subjects’ eye movements, which reveals patterns consistent with forward induction reasoning.

Large Traders, Asset Markets, and Currency Crises

During currency crises, large traders once simultaneously short the asset markets and currency market. We study the large trader's information manipulation in crises by introducing a large trader in an asset market and a currency-attack coordination game with imperfect information. The asset price realized in the asset market aggregates dispersed private information acting as a public signal in the currency attack game. We show that the incentive of the large trader to manipulate the asset price in favor of its currency attack leads to the financial contagion. In equilibrium, the large trader's manipulating the asset price to be lower and attacking the currency regime are concurrent; the large trader's manipulation in the asset market is most significant when the public signal is in the intermediate range. To draw policy implication regarding the market transparency, we show that when the asset market is transparent, a natural equilibrium refinement that incorporates forward induction reasoning would select the equilibrium where every trader behaves most aggressively in the currency-attack game and the currency regime is most fragile.

Inefficient stage Nash is not stable

It is shown that, in infinitely-repeated games between two arbitrarily patient players, strategy profiles with inefficient pure stage-Nash continuations are not strategically stable (Kohlberg and Mertens, 1986). By contrast, a set of strategy profiles similar to the Prisoners' Dilemma's “perfect tit-for-tat” is “uniformly robust to perfect entrants” (Swinkels, 1992), and hence contains a strategically stable set. Moreover, this set satisfies iterated dominance and a version of forward induction, whilst its stable subset is admissible.

Perfect forward induction

Suppose that every player in an extensive-form game incorporates perfection in conducting forward induction reasoning. To capture this idea, we propose the notion of “perfect extensive-form rationalizability” (PEFR). In every simultaneous move game, PEFR coincides with Brandenburger’s (1992) permissibility, which can be obtained by Dekel and Fudenberg’s (1990) procedure. Although PEFR is closely relate to iterated admissibility and other cautious reasoning processes, we show that there is no relationship in general.