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.
Regarding repeated games and stochastic games, Mertens 1982Mertens, Jean-François, 1982. "Repeated Games: An Overview of the Zero-sum Case," Advances in Economic Theory, edited by W. Hildenbrand, Cambridge University Press, London and New York. and 1986Mertens, Jean-François, 1986. "Repeated Games," International Congress of Mathematicians. survey articles, and his 1994Mertens, Jean-François, and Sylvain Sorin, and Shmuel Zamir, 1994. "Repeated Games," Parts A, B, C; Discussion Papers 1994020, 1994021, 1994022; Université Catholique de Louvain, Center for Operations Research and Econometrics (CORE). 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.
Epistemic models
Mertens and ZamirAn exposition for the general reader is by Shmuel Zamir, 2008: "Bayesian games: Games with incomplete information," Discussion Paper 486, Center for Rationality, Hebrew University. 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.A popular version in the form of a sequence of dreams about dreams appears in the film "Inception." The logical aspects of players' beliefs about others' beliefs is related to players' knowledge about others' knowledge; see induction puzzles for entertaining examples, and Common knowledge (logic) for another example and a precise definition.
Repeated games with incomplete information
Repeated games with incomplete information, were pioneered by Aumann and Maschler.Aumann, R. J., and Maschler, M. 1995. Repeated Games with Incomplete Information. Cambridge London: MIT Press Sorin S (2002a) A first course on zero-sum repeated games. Springer, Berlin 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.Mertens J-F (1987) Repeated games. In: Proceedings of the international congress of mathematicians, Berkeley 1986. American Mathematical Society, Providence, pp 1528–1577
(1) Information: Mertens extended the theory from the independent case where the private information of the players is generated by independent random variables, to the dependent case where correlation is allowed. (2) Signalling structures: the standard signalling theory where after each stage both players are informed of the previous moves played, was extended to deal with general signalling structure where after each stage each player gets a private signal that may depend on the moves and on the state.
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.Mertens J-F (1972) The value of two-person zero-sum repeated games: the extensive case. Int J Game Theory 1:217–227 Additionally with Shmuel Zamir,Mertens J-F, Zamir S (1971) The value of two-person zero-sum repeated games with lack of information on both sides. Int J Game Theory 1:39–64 Jean-François Mertens showed the existence of a limiting value. Such a value can be thought either as the limit of the values of the stage games, as goes to infinity, or the limit of the values of the -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.Cardaliaguet P (2007) Differential games with asymmetric information. SIAM J Control Optim 46:816–838De Meyer B. (1999), From repeated games to Brownian games, 'Annales de l'Institut Henri Poincaré, Probabilites et Statistiques', 35, 1–48. 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.Mertens J.-F. (1998), The speed of convergence in repeated games with incomplete information on one side, 'International Journal of Game Theory', 27, 343–359. 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.Mertens J.-F. and S. Zamir (1976b), The normal distribution and repeated games, 'International Journal of Game Theory', 5, 187–197. 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.'Mertens J-F, Zamir S (1976a) On a repeated game without a recursive structure. Int J Game Theory 5:173–182Sorin S (1989) On repeated games without a recursive structure: existence of . Int J Game Theory 18:45–55
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.
Stochastic games 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 1968Blackwell and Ferguson,1968. "The Big Match", Ann. Math. Statist. Volume 39, Number 1 (1968), 159–163. 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 ParthasarathyMertens, J-F., Parthasarathy, T.P. 2003. Equilibria for discounted stochastic games. In Neyman A, Sorin S, editors, Stochastic Games and Applications, Kluwer Academic Publishers, 131–172. 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.
Market games: limit price mechanism 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.
Shapley value
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 differentiability conditions have been originally required to define Shapley value of non-atomic cooperative games. But first exchanging the order of taking the "average over all possible sample sizes" and taking such a derivative, Jean-François Mertens uses the smoothing effect of such an averaging process to extend the applicability of the diagonal formula. This trick alone works well for majority games (represented by a step function applied on the percentage of population in the coalition). Exploiting even further this commutation idea of taking averages before taking derivative, Jean-François Mertens expends by looking at invariant transformations and taking averages over those, before taking the derivative. Doing so, Mertens expends the diagonal formula to a much larger space of games, defining a Shapley value at the same time.Neyman, A., 2002. Value of Games with infinitely many Players, "Handbook of Game Theory with Economic Applications," Handbook of Game Theory with Economic Applications, Elsevier, edition 1, volume 3, number 3, 00. R.J. Aumann & S. Hart (ed.).
Refinements and Mertens-stable equilibria