The Mythology of Non-Cooperative Games

Non-cooperative games study the conflict situations among players – i.e., it is the study of situations where the payoffs don’t simply depend on the player’s own action, but also on the action of another player. Game theory assumes players are rational; each player will always act to maximize their payoff given their beliefs about how other players will play.

It can be concluded that game theory is a cognitive science; thus, it has its own assumptions that players are aware of all the actions which are available to them. One of the salient features of game theory is that one can always design different payoffs, settings, number of players, and actions and information available to each player. These differences lead to different strategies, equilibria, and outcomes.¹

Research has shown that a player’s real-life behavior does not align with the predictions derived from game theory. The differences in real-life behavior and traditional assumptions of game theory can be attributed to:

• The cognitive biases of individual players while making decisions.
• A mismatch in game payoffs to players’ expected payoffs.
• A player’s limitations in thinking about other players’ strategy and behavior.

Dysfunction with Nash Equilibria

We all are human beings, so our approach to solving problems can be cognitively very different from each other. We have flexibility in our approaches, and these choices sometimes do not comply with Nash equilibrium strategies. It is not always a dysfunction; people make their choice according to their conjectures, so it is important to correct the assumptions of Nash equilibria as the use of game theory increases in various decisioning domains. According to one analysis:

“[The] Nash equilibrium (NE) concept entails the assumption that all players think in a very similar manner when assessing one another strategies. In a NE, all players in a game base their strategies not only on knowledge of the game’s structure but also on identical conjectures about what all other players will do.”²

The mythology of game theory is that these identical conjectures are applicable to all situations and settings and for all players; it is the core that is protected in game theory.¹ There are lot of works done in experimental games showing that players’ conjectures and beliefs demonstrate some deviation from the pre-assumed game theoretic mythology. An experiment was conducted that confirms this.

Experimental design and inconsistent behavior across games

A real-life experiment was carried out to confirm if human players behave as per the NE strategies.¹ A total of 180 humans participated in this experiment for two types of games: a trust game and a donation game. Each player had $5 at the start. Player 1 chooses how many dollars they want to pass to Player 2. The money passed will be increased threefold, meaning Player 2 will receive the tripled amount (plus their original $5). Player 2 decides how much they will return to Player 1. This information is available to all the players, and their choices are private and anonymous with respect to other players. The subgame perfect Nash equilibrium (SPNE) is that either player will pass and return $0 as the dominant strategy equilibrium. The assumption is always that each player wants to maximize their payoffs and each player believes that the other player will also do the same. That leads to the thought “Player 2 will return $0; I will not pass any money to them”. This is the equilibrium strategy assumed in Nash beliefs.

Standard approaches that show deviations from NE strategies assumes players will not follow game theoretical expectations. Across both games, inconsistency was observed in the players’ behavior. In a trust game, it was observed that 56% of participants as Player 1 sent money ($1.43 on average); as Player 2, they returned $1.23 on average. These deviations cast doubts on Nash beliefs. Out of 100 players who received money as Player 2, only 62 of them returned on average $2.22. Those 62 who returned money were not consistent in sending money. There are 60 players who behaved consistently according to the Nash equilibrium in both trust game roles but lacked consistency in the donation game.

Are beliefs and behavior consistent?

Various reasons from cognitive science force us to doubt the behavior and beliefs of the players across different game settings. The human mind is always shifting and changing its way of thinking, which explains why players are not consistent in their choices. It is also evident from this experiment that players show variance in their choices while playing games.
The mythology of game theory assumes that players in a game environment have their own preference and payoff maximization strategy and know how other players will act and think. In this mythology, we cannot chalk out the difference if someone asks the players to select a particular action in the game before making predictions or vice versa. The results also suggest that changing the belief and order of choice affects players’ beliefs and this change in task does not comply with Nash equilibrium.

Conclusion

The results showed that players mostly deviate from NE beliefs. The deviations observed are not simple, consistent, and easy to explain; they keep on changing as per the environment. This experiment also proved that players do not have shared beliefs. Rather, their own beliefs seem to be fit for one game environment and can change in another environment. So, it may be misleading to conclude these results are deviations from Nash beliefs. Equilibrium concepts are not based on how humans think or make decisions in real life, so models that use false assumptions may not create problems when prediction is the only goal (as opposed to understanding).
However, Nash equilibrium models failed to predict real-life behavior, so we should not accept the success of these models in prediction scenario due to use of false assumptions. Rather than trying to fit humans in an old, developed mythology, we should build models where we can fit humans with their real behavior. No model has been developed till now, but research is ongoing.

References

1. McCubbins, Mathew D., Mark Turner, and Nicholas Weller. “The Mythology of Game Theory” in Proceedings of the International Conference on Social Computing, Behavioral-Cultural Modeling & Prediction 1-8. (2012)
2. Lupia, A., A. S. Levine, and N. Zharinova. “Should Political Scientists Use the Self Confirming Equilibrium Concept? Benefits, Costs, and an Application to the Jury Theorem.” Political Analysis 18:103-123. (2010)

Authored by:  Akash Omer, Senior Consultant AI/ML at Absolutdata