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From game theory to cognitive hierarchy theory

| Updated: January 03, 2020 20:35:51

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In recent times, game theory implications of different issues have been coming up in the news quite frequently. Starting from Brexit to Rohingya issue, it seems like everyone is turning to game theory for answers. This is probably because, game theory provides us with tools that enable us to make predictions about strategic interactions. When outcomes for actors depend on the actions of each other it is called a situation of strategic interaction and the study of strategic interaction is referred to as the game theory in Economics. In the past much economic analysis ignored strategic interactions by assuming that economic actors interacted through markets and each actor made up only a very small part of the market. Prior to 1900 there were few examples where interactions of a finite number of actors were studied. The development of game theory in the 20th century, led by the publication of a book by von Neumann and Morgenstern (1944), provided a formal structure for the analysis of strategic interactions.

For Economists the word game implies a process of interaction with a prescribed population of participants, a game has a set of rules and a set of payoffs associated to every possible outcome of the game. A game can be described using three elements. They are: a set of players; a set of strategies available to each player and a payoff function that, given the input of a chosen strategy for each player, specifies the outcome of the game for each of the players. We know that in strategic interactions, players may have a dominant strategy or a dominated strategy based on their prediction of the behaviour of other players. Nevertheless, sometimes it might be the case that there is no dominated strategy. However, in almost all games there will be at least one stable strategy or the Nash equilibrium. As a result, game theory has been generally regarded as a predictive and descriptive tool of human behaviour under certain conditions. However, strength of game theory in predicting human behaviour is heavily reliant on various notions of rationality and mutual knowledge of each other's rationality. Thus, anyone willing to adopt game theory in order to understand human behaviour should also be mindful about the assumptions of rationality that they set.

In recent years a new stream of literature has emerged taking essence from both behavioural economics and game theory. One of such behavioural model, the cognitive hierarchy theory attempts to describe human thought processes in strategic games. This new model, also known as Level-k thinking aims to improve upon the accuracy of predictions made by standard analytic methods. Level-k framework assumes that players in strategic games base their decisions on their predictions about the likely actions of other players. According to level-k, players in strategic games can be categorised by the 'depth' of their strategic thought. It is thus heavily focused on bounded rationality. In its basic form, level-k theory implies that each player believes that he or she is the most sophisticated person in the game. Therefore, players at some level k will neglect the fact that other players could also be level-k, or even higher.

However, a player can also acknowledge the fact that there could be different levels of players in the game and thus, iterate their decisions accordingly. A typical example of level-k thinking is the use of Keynesian Beauty Contest. It was first developed by John Maynard Keynes and introduced in The General Theory of Employment, Interest and Money (1936), to explain price fluctuations in equity markets. In the Keynesian beauty contest, participants are asked to choose a number that will be as close as possible to some fraction of the average of all participants' guesses. Now, if a player believes that half of the players are level-zero and half are level-one, then he would select a number about halfway between the guesses of the typical level-one and level-two players. Suppose there are many players, each attempting to guess half of the average from the range 1-100. A level-zero player will select a number non-strategically. That number might be selected at random. A level-one player will choose the number consistent with the belief that all other players are level-zero. If all other players in the game are level-zero, the average of those guesses would be about 50. Therefore, a level-one player will choose 25. A level-two player will choose the number consistent with the belief that all other players are level-one. Since a level-one player will choose 25, a level-two player should choose 13. This process repeats for higher-level players.

Theories of behaviour often assume that players think strategically, meaning that players will base their actions on the probable decisions of other players in a way that will serve their objectives. Yet, many games, both real and contrived, do not result in the equilibrium predicted by standard analytic methods. The standard solution to the Keynesian Beauty Contest is determined by iterated elimination of dominated strategies. Using the example above, a fully rational player will observe that the number could be 50. This player will also predict that the other players know that as well and will behave accordingly, so the maximum feasible number becomes 25. But, again, other players should know that too. This process repeats indefinitely, and concludes with all players selecting 0, the Nash equilibrium for this game. Nevertheless, a number of theorists including: Nagel (1995) and Camerar et al. (2004) have shown that most players choose numbers around either 25 or 13. These guesses are consistent with the first and second order depth of reasoning of the cognitive hierarchy theory and the findings of their experiments suggest that only a small proportion of players' exhibit depths of reasoning greater than the second order.

Although, many alternative models have been proposed to explain the divergences between standard theory and experimental results, they have been mostly non-predictive. In this regard, cognitive hierarchy theory explains the observed pattern of opportunistic cooperation found in many games and provides a better explanation of human behaviour. Furthermore, cognitive hierarchy theory can also be incorporated into existing models because researchers are able to preserve the common assumption that players are self-interested. Experts now suggest that cognitive hierarchy theory can offer reasonably accurate predictions about human behaviour while acknowledging stronger forms of bounded rationality and opportunism than standard theory. Consequently, policy makers and diplomates alike can now benefit to a great extent from the insights of cognitive hierarchical theory in place of standard game theory.

Rafiqua Ferdousi  is Research Economist, SANEM.


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