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Many Agent Games in Socio-economic Systems: Corruption, Inspection, Coalition Building, Network Growth, Security
Analysis of games with many players lies in the centre of the modern game theory. The general picture of game theoretic modelling dealt with in our book is characterized by a set of big players, also referred to as principals or major agents, acting on the background of large pools of small players. The impact of the behaviour of each small player in a group on the overall evolution decreasing with the increase of the size of the group.
Two approaches to the analysis of such systems are clearly distinguished and are dealt with in Parts I and II. In the first approach, players in groups are not independent rational optimizers. They are either directly controlled by principals and serve the interests of the latter (pressure and collaboration setting) or they resist the actions of the principals (pressure and resistance setting) by evolving their strategies in an 'evolutionary manner' via interactions with other players subject to certain clear rules, deterministic or stochastic. Such interactions, often referred to as myopic or imitating, include the exchange of opinions or experience, with some given probabilities of moving to strategies that are more profitable. They can also evolve via the influence of social norms. The examples of the real world problems involved include government representatives (often referred to in the literature as benevolent dictators) chasing corrupted bureaucrats, inspectors chasing tax-paying avoidance, police acting against terrorist groups or models describing the attacks of computer or biological viruses. This includes the problem of optimal allocation of the budget or efforts of the big player to different strategies affecting small players, for instance, the allocation of funds (corrected in real time) for the financial support of various business or research projects. Other class of examples concerns appropriate (or better optimal) management of complex stochastic systems consisting of large number of interacting components (agents, mechanisms, vehicles, subsidiaries, species, police units, robot swarms, etc.), which may have competitive or common interests. Such management can also deal with the processes of merging and splitting of functional units (say, firms or banks) or the coalition building of agents. The actions of the big players effectively control the distribution of small players among their possible strategies and can influence the rules of their interaction. Several big players can also compete for more effective pressure on small players. This includes, in particular, the controlled extensions of general (nonlinear) evolutionary games. Under our approach the classical games of evolutionary biology, like hawk and dove game, can be recast as a controlled process of the propagation of violence, say in regions with mixed cultural and/or religious traditions. For discrete state spaces, V. Kolokoltsov introduced the games of this kind under the name of nonlinear Markov games.
In the second approach, small players in groups are themselves assumed to be rational optimizers, though in the limit of large number of players the influence of the decisions of each individual player on the whole evolution becomes negligible. The games of this type are referred to as mean field games. They were introduced about 15 years ago by two research groups in France and Canada, and since then developed into one of the most active directions of research in game theory. We shall discuss this setting mostly in combination with the first approach, with evolutionary interactions (more precisely, pressure and resistance framework) and individual decision-making taken into account simultaneously. This combination leads naturally to two-dimensional arrays of possible states of individual players, one dimension controlled by the principals and/or evolutionary interactions and another dimension by individual decisions.
Carrying out a traditional Markov decision analysis for a large state space (large number of players and particles) is often unfeasible. The general idea for our analysis is that under rather general assumptions, the limiting problem for a large number of agents can be described by a well manageable deterministic evolution, which represents a performance of the dynamic law of large numbers (LLN). This procedure turns the 'curse of dimensionality' to the 'blessing of dimensionality'. As we show all basic criteria of optimal decision making (including competitive control) can be transferred from the models of a large number of players to a simpler limiting decision problem. Since even the deterministic limit of the combined rational decision making processes and evolutionary type models can become extremely complex, another key idea of our analysis is in searching for certain reasonable asymptotic regimes, where explicit calculations can be performed. Several such regimes are identified and effectively used. We deal mostly with discrete models, thus avoiding technicalities arising in general models (requiring stochastic differential equations or infinite-dimensional analysis). Extensions dealing with general jump-type Markov processes are often straightforward.
From the practical point of view, the approaches developed here are mostly appropriate for dealing with socioeconomic processes that are not too far from an equilibrium. For those processes, the equilibria play the role of the so-called turnpikes, that is, attracting stationary developments. Therefore, much attention in our research is given to equilibria (rest points of the dynamics) and their structural and dynamic stability. For processes far from equilibria other approaches seem to be more relevant, for instance the methods for the analysis of turbulence. Another problem needed to be addressed for concrete applications of our models lies in the necessity to get hold of the basic parameters entering its formulation, which may not be that easy. Nevertheless, the strong point of our approach is that it requires identifying really just a few real numbers (not multi-dimensional distributions), which may be derived in principle from statistical experiments or field observations.
