2:30pm - 3:45pm
Wednesday 18 March 2020

CANCELLED - Economics Seminar: Why Macroeconomic Game Theory Needs Monte Carlo Integration


Prof. Peter Hammond (University of Warwick) is visiting the School. Peter's reserach interests include games and allocation mechanisms with many agents, social choice and individual well-being, gains from economics migration, ethics and economic policy, players' beliefs in experimental games and quantum probabilty among others.


Room: 40 AD 00
University of Surrey
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Why Macroeconomic Game Theory Needs Monte Carlo Integration


Much of modern macroeconomic theory involves games of incomplete information between: (i) a continuum of agents represented by points of the Lebesgue unit interval; (ii) a principal whose policy choice influences economic outcomes. Typically this interaction is modelled as an aggregative game where: (i) each agent's payoff is some function whose three arguments are that agent's own action, the principal's action, and some aggregate of all the agents' actions; (ii) the principal's payoff is some function of the principal's own action and of the same aggregate of all the agents' actions. When agents have independently distributed random types, the typical associated strategy profile for the agents, which maps type profiles into action profiles, is described by a continuum of agents' independent random action choices.

A fundamental difficulty with this formulation is that the associated sample path is only Lebesgue measurable in degenerate cases when agents' types are essentially deterministic. This non-measurability implies that the aggregate specified in the principal's and each agent's payoff function can only be represented as a Lebesgue integral in degenerate cases. In particular, the mean of all agents' action choices is almost surely undefined.

To overcome this difficulty, this paper applies the idea behind the Monte Carlo approach to approximating an integral numerically. First one takes a large random sample of points in the domain that are independently and identically drawn from a uniform distribution. Then one calculates the sample average of the associated values of the integrand, and considers the almost sure limit as the sample size tends to infinity.

We apply the Monte Carlo approach to a continuum of independent random variables rather than to a measurable function. Assuming that the mapping from the continuum to the distribution of each random variable is measurable, it follows from Kolmogorov's strong law of large numbers that this "Monte Carlo integral" is well defined as a random variable. Moreover, its limiting distribution is that of a degenerate random variable which almost surely equals the Lebesgue integral of the different expected values.

Finally, the Aumann integral of a measurable correspondence is defined as the range of Lebesgue integrals of measurable selections from the correspondence. One can also define the "Monte Carlo Aumann integral" of a correspondence as the range of Monte Carlo integrals of independent random selections, even though these selections  are typically not measurable. A trivial implication of this definition is that the range of this integral is always a convex set, even in infinite dimensional linear spaces where the Aumann integral may not be convex. Provided that payoff functions are continuous, this should allow one to prove that a Monte Carlo integral version of Bayesian--Nash equilibrium in pure strategies exists for infinite dimensional compact and convex action spaces in which a suitable version of Kakutani's fixed point theorem holds.

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