Tarski’s Theorem, Supermodular Games, and the Complexity of Equilibria

The use of monotonicity and Tarski's theorem in existence proofs of equilibria is very widespread in economics, while Tarski's theorem is also often used for similar purposes in the context of verification. However, there has been relatively little in the way of analysis of the complexity of finding the fixed points and equilibria guaranteed by this result. We study a computational formalism based on monotone functions on the $d$-dimensional grid with sides of length $N$, and their fixed points, as well as the closely connected subject of supermodular games and their equilibria. It is known that finding some (any) fixed point of a monotone function can be done in time $\log^d N$, and we show it requires at least $\log^2 N$ function evaluations already on the 2-dimensional grid, even for randomized algorithms. We show that the general Tarski problem of finding some fixed point, when the monotone function is given succinctly (by a boolean circuit), is in the class PLS of problems solvable by local search and, rather surprisingly, also in the class PPAD. Finding the greatest or least fixed point guaranteed by Tarski's theorem, however, requires $d\cdot N$ steps, and is NP-hard in the white box model. For supermodular games, we show that finding an equilibrium in such games is essentially computationally equivalent to the Tarski problem, and finding the maximum or minimum equilibrium is similarly harder. Interestingly, two-player supermodular games where the strategy space of one player is one-dimensional can be solved in $O(\log N)$ steps. We also observe that computing (approximating) the value of Condon's (Shapley's) stochastic games reduces to the Tarski problem. An important open problem highlighted by this work is proving a $\Omega(\log^d N)$ lower bound for small fixed dimension $d \geq 3$

A theory of dynamic tariff and quota retaliation

This paper establishes relationships between static Nash equilibria and dynamic Markov perfect equilibria of tariff and quota retaliation games. In supermodular games where tariffs are strategic complements, the steady state of every, symmetric Markov perfect equilibrium must have lower tariffs than in the static equilibrium. If tariffs are strategic substitutes, tariffs in the dynamic game are higher than in the static equilibrium. The supermodular case is extended to quota competition. Instead of the well-known non-equivalence between tariff and quota retaliation outcomes under complete myopia, in some circumstances, free trade can be supported in the steady state of a Markov perfect equilibrium, regardless of whether policies employed are quotas or tariffs. We reach the conclusion that the effect of introducing dynamics crucially depends on whether the policy instruments employed by the countries are strategic substitutes or complements irrespective of whether they are tariffs or quotas.Foreign trade policy; Tariff; Quota; Retaliation; Dynamic Game; Markov perfect equilibrium; Supermodular games

A Theory of Dynamic Tariff and Quota Retaliation

This paper characterizes, under the most general conditions to date, the steady-state equilibria of a symmetric, two-country trade model in which countries move in alternating-move, dynamic either tariffsetting or quota-setting games in Markov Perfect strategies, and compares the respective equilibrium level of tariffs and quotas with the corresponding pairs in the equilibria of static games. Our results imply that the alleged non-equivalence of the outcomes of tariff-retaliation (neither free trade nor autarky) and quota-retaliation (asymptotic autarky) games in the literature depends crucially on complete myopia, and can be dismissed altogether once dynamic considerations are introduced in an operationally significant manner.Foreign trade policy; Tariff; Quota; Retaliation; Dynamic Game; Markov perfect equilibrium; Supermodular games

Dynamic Games under Bounded Rationality

I propose a dynamic game model that is consistent with the paradigm of bounded rationality. Its main advantages over the traditional approach based on perfect rationality are that: (1) under given state the strategy space is a chain-complete partially ordered set; (2) the response function satisfies certain order-theoretic property; (3) the evolution of economic system is described by the Dynamical System defined by iterations of the response function; (4) the existence of equilibrium is guaranteed by fixed point theorems for ordered structures. If the preference happens to be represented by a utility function and the response was derived from utility maximization, then the equilibrium defined by fixed points of the response function will be the same as Nash equilibrium. This preference-response framework liberates economics from the utility concept, and constitutes a synthesis between normal-form and extensive-form games. And the essential advantages of our preference-response approach was secured by successfully resolving some long-standing paradoxes in classical theory, yielding straightforward ways out of the impossibility theorem of Arrow and Sen, the Keynesian beauty contest, the Bertrand Paradox, and the backward induction paradox. These applications have certain characteristics in common: they all involve important modifications in the concept of perfect rationality

A theory of dynamic tariff and quota retaliation

This paper establishes relationships between static Nash equilibria and dynamic Markov perfect equilibria of tariff and quota retaliation games. In supermodular games where tariffs are strategic complements, the steady state of every, symmetric Markov perfect equilibrium must have lower tariffs than in the static equilibrium. If tariffs are strategic substitutes, tariffs in the dynamic game are higher than in the static equilibrium. The supermodular case is extended to quota competition. Instead of the well-known non-equivalence between tariff and quota retaliation outcomes under complete myopia, in some circumstances, free trade can be supported in the steady state of a Markov perfect equilibrium, regardless of whether policies employed are quotas or tariffs. We reach the conclusion that the effect of introducing dynamics crucially depends on whether the policy instruments employed by the countries are strategic substitutes or complements irrespective of whether they are tariffs or quotas

A theory of dynamic tariff and quota retaliation

This paper establishes relationships between static Nash equilibria and dynamic Markov perfect equilibria of tariff and quota retaliation games. In supermodular games where tariffs are strategic complements, the steady state of every, symmetric Markov perfect equilibrium must have lower tariffs than in the static equilibrium. If tariffs are strategic substitutes, tariffs in the dynamic game are higher than in the static equilibrium. The supermodular case is extended to quota competition. Instead of the well-known non-equivalence between tariff and quota retaliation outcomes under complete myopia, in some circumstances, free trade can be supported in the steady state of a Markov perfect equilibrium, regardless of whether policies employed are quotas or tariffs. We reach the conclusion that the effect of introducing dynamics crucially depends on whether the policy instruments employed by the countries are strategic substitutes or complements irrespective of whether they are tariffs or quotas

Understanding economic decision-making under social norms prescribing behaviours

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Mitigation of flood risks : the economic problem

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