Jean-François Mertens

Jean-François Mertens (11 March 1946 – 17 July 2012) was a Belgian game theorist and mathematical economist.

Mertens contributed to economic theory in regards to order-book of market games, cooperative games, noncooperative games, repeated games, epistemic models of strategic behavior, and refinements of Nash equilibrium (see solution concept).

In cooperative game theory he contributed to the solution concepts called the core and the Shapley value.

Stochastic games were introduced by Lloyd Shapley in 1953.

The first paper studied the discounted two-person zero-sum stochastic game with finitely many states and actions and demonstrates the existence of a value and stationary optimal strategies.

The study of the undiscounted case evolved in the following three decades, with solutions of special cases by Blackwell and Ferguson in 1968 and Kohlberg in 1974.

The existence of an undiscounted value in a very strong sense, both a uniform value and a limiting average value, was proved in 1981 by Jean-François Mertens and Abraham Neyman.

The study of the non-zero-sum with a general state and action spaces attracted much attention, and Mertens and Parthasarathy proved a general existence result under the condition that the transitions, as a function of the state and actions, are norm continuous in the actions.

Mertens had the idea to use linear competitive economies as an order book (trading) to model limit orders and generalize double auctions to a multivariate set up.

Acceptable relative prices of players are conveyed by their linear preferences, money can be one of the goods and it is ok for agents to have positive marginal utility for money in this case (after all agents are really just orders!).

In fact this is the case for most order in practice.

More than one order (and corresponding order-agent) can come from same actual agent.

In equilibrium good sold must have been at a relative price compared to the good bought no less than the one implied by the utility function.

Goods brought to the market (quantities in the order) are conveyed by initial endowments.

Limit order are represented as follows: the order-agent brings one good to the market and has non zero marginal utilities in that good and another one (money or numeraire).

An at market sell order will have a zero utility for the good sold at market and positive for money or the numeraire.

Mertens clears orders creating a matching engine by using the competitive equilibrium – in spite of most usual interiority conditions being violated for the auxiliary linear economy.

Mertens's mechanism provides a generalization of Shapley–Shubik trading posts and has the potential of a real life implementation with limit orders across markets rather than with just one specialist in one market.

The diagonal formula in the theory of non-atomic cooperatives games elegantly attributes the Shapley value of each infinitesimal player as his marginal contribution to the worth of a perfect sample of the population of players when averaged over all possible sample sizes.

Such a marginal contribution has been most easily expressed in the form of a derivative—leading to the diagonal formula formulated by Aumann and Shapley.

This is the historical reason why some

Regarding repeated games and stochastic games, Mertens 1982 and 1986 survey articles, and his 1994 survey co-authored with Sylvain Sorin and Shmuel Zamir, are compendiums of results on this topic, including his own contributions.

Mertens also made contributions to probability theory and published articles on elementary topology.

Mertens and Zamir implemented John Harsanyi's proposal to model games with incomplete information by supposing that each player is characterized by a privately known type that describes his feasible strategies and payoffs as well as a probability distribution over other players' types.

They constructed a universal space of types in which, subject to specified consistency conditions, each type corresponds to the infinite hierarchy of his probabilistic beliefs about others' probabilistic beliefs.

They also showed that any subspace can be approximated arbitrarily closely by a finite subspace, which is the usual tactic in applications.

Repeated games with incomplete information, were pioneered by Aumann and Maschler.

Two of Jean-François Mertens's contributions to the field are the extensions of repeated two person zero-sum games with incomplete information on both sides for both (1) the type of information available to players and (2) the signalling structure.

In those set-ups Jean-François Mertens provided an extension of the characterization of the minmax and maxmin value for the infinite game in the dependent case with state independent signals.

Additionally with Shmuel Zamir, Jean-François Mertens showed the existence of a limiting value.

Such a value can be thought either as the limit of the values v_n of the n stage games, as n goes to infinity, or the limit of the values v_{\lambda} of the {\lambda}-discounted games, as agents become more patient and.

A building block of Mertens and Zamir's approach is the construction of an operator, now simply referred to as the MZ operator in the field in their honor.

In continuous time (differential games with incomplete information), the MZ operator becomes an infinitesimal operator at the core of the theory of such games.

Unique solution of a pair of functional equations,

Mertens and Zamir showed that the limit value may be a transcendental function unlike the maxmin or the minmax (value in the complete information case).

Mertens also found the exact rate of convergence in the case of game with incomplete information on one side and general signalling structure.

A detailed analysis of the speed of convergence of the n-stage game (finitely repeated) value to its limit has profound links to the central limit theorem and the normal law, as well as the maximal variation of bounded martingales.

Attacking the study of the difficult case of games with state dependent signals and without recursive structure, Mertens and Zamir introduced new tools on the introduction based on an auxiliary game, reducing down the set of strategies to a core that is 'statistically sufficient.'

Collectively Jean-François Mertens's contributions with Zamir (and also with Sorin) provide the foundation for a general theory for two person zero sum repeated games that encompasses stochastic and incomplete information aspects and where concepts of wide relevance are deployed as for example reputation, bounds on rational levels for the payoffs, but also tools like splitting lemma, signalling and approachability.

While in many ways Mertens's work here goes back to the von Neumann original roots of game theory with a zero-sum two person set up, vitality and innovations with wider application have been pervasive.