An Alternative to Nash Equilibrium
Every decision we make is based on some combination of past experience, knowledge of how the world works, and our own personal values and preferences. While some decisions are easy, most of the decisions we make are the results of our best guesses at what might give us the largest reward. Game theory tries to explain these decisions, assuming people are rational and driven by what they perceive is best for them.
Most situations we are put in can be modeled as “games” in which there are predefined rules and a set of possible outcomes that can be reached based on the choices of all participants in the game. In any of these games, there exist “nash equilibria”, in which no player has incentive to change their strategy. It seems intuitive that games would tend towards a state of nash equilibrium, since nash equilibrium is reached when players adopt strategies of mutual best responses, but in real life, it’s not this simple. Two researchers, Yakov Babichenko and Aviad Rubenstein, recently demonstrated that many games in the real world never reach nash equilibria, and there is no generalizable way for participants of a game to adapt their strategies to guarantee reaching a nash equilibrium without knowing all relevant information about the game and other participants in the game, which is impossible in most real-life situations. In other words, all games have nash equilibria, but we can’t assume that all games will be played at their equilibria.
A similar concept to nash equilibrium is correlated equilibrium. In correlated equilibrium, players make decisions based on a “correlating” signal. If all players have no incentive to choose a strategy different from the one indicated by the signal, then the game they are part of is said to be in correlated equilibrium. A classic example of correlated equilibrium is when sports teams decide on who starts based on a coin flip. Correlated equilibria can include all sets of actions given by nash equilibria as well as ones not described by nash equilibria. Because nash equilibria are not by any means the optimal states for games, often correlated equilibria can correspond to game results that are better for everyone than the nash equilibria. Furthermore, researchers have found that in many complex games where reaching a nash equilibrium is hard, a correlated equilibrium is often more natural. For example, when researchers modeled games with players who took a regret minimization approach, the games often quickly converged to a correlated equilibrium. A researcher at MIT, Constantinos Daskalakis, commented that it was as if “the [correlating] device was somehow implicitly found, through the interaction.”
While finding a correlated equilibrium is a natural and intuitive way to reach a stable game state that maximizes the total reward players collectively get out of a game (a correlating device tells players how they should act), it is unrealistic to expect that a large percentage of our decisions can be made based on a correlating device of some sort, and that correlated equilibria can even be found for some complex games with many players. Correlated equilibrium expands our understanding of how games are played in the real world and how we should reach states of the highest mutual benefit in certain situations, but it is still not generalizable to all games. At the end of the day, even we don’t always know what we want and what’s best for ourselves and besides, our desires and values are constantly changing. We can generate great models for games and the decision making processes that go into them, but there will always be some games that we can’t solve.
Sources:
https://arxiv.org/abs/1608.06580
https://www.quantamagazine.org/in-game-theory-no-clear-path-to-equilibrium-20170718/