48 Game Theory
Vintage game theorists is the best kind of economist (and some of the macroeconomists do even sometimes interpret their models as mean-field games) (Beatrice Cherrier)
48.1 Nash Equilibrium
Investopedia
Nash equilibrium is a concept within game theory where the optimal outcome of a game is where there is no incentive to deviate from the initial strategy. More specifically, the Nash equilibrium is a concept of game theory where the optimal outcome of a game is one where no player has an incentive to deviate from their chosen strategy after considering an opponent’s choice.1
Overall, an individual can receive no incremental benefit from changing actions, assuming other players remain constant in their strategies. A game may have multiple Nash equilibria or none at all.
Nash equilibrium is often compared alongside dominant strategy, both being strategies of game theory. The Nash equilibrium states that the optimal strategy for an actor is to stay the course of their initial strategy while knowing the opponent’s strategy and that all players maintain the same strategy, as long as all other players do not change their strategy.
Dominant strategy asserts that the chosen strategy of an actor will lead to better results out of all the possible strategies that can be used, regardless of the strategy that the opponent uses.
All models of game theory only work if the players involved are “rational agents,” meaning that they desire specific outcomes, operate in attempting to choose the most optimal outcome, incorporate uncertainty in their decisions, and are realistic in their options.
Both the terms are similar but slightly different. Nash equilibrium states that nothing is gained if any of the players change their strategy if all other players maintain their strategy. Dominant strategy asserts that a player will choose a strategy that will lead to the best outcome regardless of the strategies that other plays have chosen. Dominant strategy can be included in Nash equilibrium whereas a Nash equilibrium may not be the best strategy in a game.
The prisoner’s dilemma is a common situation analyzed in game theory that can employ the Nash equilibrium. In this game, two criminals are arrested and each is held in solitary confinement with no means of communicating with the other. The prosecutors do not have the evidence to convict the pair, so they offer each prisoner the opportunity to either betray the other by testifying that the other committed the crime or cooperate by remaining silent.
If both prisoners betray each other, each serves five years in prison. If A betrays B but B remains silent, prisoner A is set free and prisoner B serves 10 years in prison or vice versa. If each remains silent, then each serves just one year in prison.
The Nash equilibrium in this example is for both players to betray each other. Even though mutual cooperation leads to a better outcome if one prisoner chooses mutual cooperation and the other does not, one prisoner’s outcome is worse.
The primary limitation of the Nash equilibrium is that it requires an individual to know their opponent’s strategy. A Nash equilibrium can only occur if a player chooses to remain with their current strategy if they know their opponent’s strategy.
48.2 Mean-Field Game Theory (MFG)
Kaust
In the theory of mean-field games (MFG), the concept of Nash equilibrium and the rational expectation hypothesis are combined to produce mathematical models for large systems, with infinitely many indistinguishable rational players.
While not all economists agree that the behavior of agents can be reduced to a precise mathematical formulation, utility maximization principles and game-theoretical equilibria explain, at least partially, many economic phenomena.
The concept of competing agents is illustrated in the works of A. Cournot and L. Walras. Indeed, Walras, also known as the founder of the École de Lausanne, refers to Cournot in 1873, as the first au- thor to seriously recur to the mathematical formalism in investigating economic problems. Cournot duopoly model is one of the earliest for- mulations of a non-cooperative game.
The term indistinguishable refers to a setting where agents share common structures of the model, though they are allowed to have heterogeneous states. In other terms, the MFG theory enables us to investigate the solution concept of Nash equilibrium, for a large population of heterogeneous agents, under the hypothesis of rational expectations.
The mean-field game formalism was developed in a series of semi- nal papers by J.-M. Lasry and P.-L. Lions [118, 119, 120] and M. Huang, R. Malhamé and P. Caines [103, 106]. It comprises methods and techniques to study differential games with a large population of rational players. These agents have preferences not only about their state (e.g., wealth, capital) but also on the distribution of the remain- ing individuals in the population. Mean-field games theory studies generalized Nash equilibria for these systems. Typically, these mod- els are formulated in terms of partial differential equations, namely a transport or Fokker-Plank equation for the distribution of the agents coupled with a Hamilton-Jacobi equation.
The mean-field game formalism was developed in a series of semi- nal papers by J.-M. Lasry and P.-L. Lions [118, 119, 120] and M. Huang, R. Malhamé and P. Caines [103, 106]. It comprises methods and techniques to study differential games with a large population of rational players. These agents have preferences not only about their state (e.g., wealth, capital) but also on the distribution of the remain- ing individuals in the population. Mean-field games theory studies generalized Nash equilibria for these systems. Typically, these mod- els are formulated in terms of partial differential equations, namely a transport or Fokker-Plank equation for the distribution of the agents coupled with a Hamilton-Jacobi equation.
MFG methods are equilibrium models where all agents are rational. In simple problems, fundamental questions such as unique- ness, existence or stability were investigated extensively. However, many MFG problems arising in mathematical economics raise issues that cannot be dealt with the current results.
In this book, we have attempted to illustrate the main techniques and methods in mean-field games theory, in several simplified models motivated by economic considerations.
48.3 Mechanism Design
Investopedia
Mechanism design is a branch of microeconomics that explores how businesses and institutions can achieve desirable social or economic outcomes given the constraints of individuals’ self-interest and incomplete information. When individuals act in their own self-interest, they may not be motivated to provide accurate information, creating principal-agent problems.
Mechanism design theory generally takes a reverse approach to game theory. It studies a scenario by beginning with an outcome and understanding how entities work together to achieve a particular outcome.
Both game theory and design theory look at the competing and cooperative influences of entities in the process towards an outcome. Mechanism design theory considers a particular outcome and what is done to achieve it. Game theory looks at how entities can potentially influence several outcomes.
Nobel 2007
48.4 Market Design
Roth Abstract
We interview each other about how game theory and mechanism design evolved into practical market design. When we learned game theory, games were modeled either in terms of the strategies available to the players (“noncooperative games”) or the outcomes attainable by coalitions (“cooperative games”), and these were viewed as models for different kinds of games. The model itself was viewed as a mathematical object that could be examined in its entirety. Market design, however, has come to view these models as complementary approaches for examining different ways marketplaces operate within their economic environment. Because that environment can be complex, there will be unobservable aspects of the game. Mathematical models themselves play a less heroic, stand-alone role in market design than in the theoretical mechanism design literature. Other kinds of investigation, communication, and persuasion are important in crafting a workable design and helping it to be adopted, implemented, maintained, and adapted.