Level-k theory is a human behavioral model used in behavioral economics and game theory. It assumes that players in strategic games base their decisions on predictions about the actions of other players. Players can thus be categorized by the depth of their strategic thought. Level-k theory assumes that each player believes that he or she is the most sophisticated person in the game. This has been attributed to many factors, such as “maintenance costs” or simply overconfidence.
Players are thus given a classification within level k based on the level of sophistication that a player ascribes to the other competitors. The more naive you assume them to be the lower your classification. This pattern continues for higher-level players, but each player has only a finite depth of reasoning, meaning that individual players have a limit to the depth to which they can reason strategically.
Economic theory, in particular behavioral economics and game theory, involves the modeling of human behavior. Human decision makers make choices that respond to economic incentives and the structure of the game (the strategic environment). The mapping between the game and the actions taken in that game involve a behavioral model. Level-k theory is a leading behavioral model in the behavioral economics literature.
Behavioral models in economics are often based on the principles of bounded rationality. These principles often imply that humans do not possess infinite computational powers but instead take mental shortcuts or use heuristics. One of these heuristics, in the present context, is an iterative model, where an individual begins with a naive belief, computes the best response to that belief, the best response to that best response, and so on.
Level-k theory emerged in the writings of Dale Stahl and Wilson. Stahl  proposed that “player types are drawn from a hierarchy of smartness analogous to the levels of iterated rationalizability.”
The hierarchy begins with some very naive type. This completely non-strategic “level-zero” player will choose actions without regard to the actions of other players. Such a player is said to have zero-order beliefs.
A one level higher sophisticated type believe the population consists of all naive types. This slightly more sophisticated (the level one) player believes that the other players will act non-strategically; his or her action will be the best response consistent with those first-order beliefs.
The next level believes the population consists of the first level. This more sophisticated (level two) player acts on the belief that the other players are level one. This pattern continues for higher-level players, but each player has only a finite depth of reasoning, meaning that individual players have a limit to the depth to which they can reason strategically.
As a motivating example, consider the “guessing game” investigated in Nagel . In that game, players simultaneously state a number between 0 and 100. The player who is closest to 2/3 of the average wins a prize. A person is defined to be of hierarchy level n if he chooses 50(2/3)^n. So a level-1 should choose 33.33, a level-2 is 22.2, etc. The optimal choice in the Nagel  experiments, given the observed empirical frequency, was 25, corresponding to about level-2. Nagel  found that the largest modes were at level-1 and level-2 choices, with a much smaller mode at level-3 and very little past that.
Hierarchical bounded rationality is a class of models of heterogeneous decision makers with different iterative reasoning models of others beginning with some naïve belief — also known as a “prior of insufficient reason.” The type of player holding this naïve belief is known as the naïve type — also called level-1. The naïve belief of this level-1 type is that the population of other players consists of individuals — collectively called level-0 players — who draw their actions from a distribution corresponding to this prior of insufficient reason. These level-0 players do not best respond to a belief, although they may react to past history and/or the game payoffs. The prior of insufficient reason could be a belief that the level-0 subpopulation population is uniformly distributed over all feasible actions  Alternatively, it could be a belief that the population is distributed over nondominated actions or with a tilt away from dominated actions, or that the population comes from a distribution derived from what was observed in the past.
To accommodate that theory using econometric methods, Stahl and collaborators  proposed a Mixture Model. A mixture model is an econometric approach wherein sub-populations exist representing each type, and these subpopulations can be identified in some proportions.
Level-k theory 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 categorized 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. Players at some level k will neglect the fact that other players could also be level-k, or even higher. This has been attributed to many factors, such as “maintenance costs” or simply overconfidence.
An important element of the econometric modeling of level-k is the distribution of choices within each level-k type. Players within each type make choices that don’t conform precisely to the prescribed behavior corresponding to their type. The degree and pattern of deviation from their prescribed choices determines the classification of a player as one type or another. Alternative behavioral econometric models for the characterization of player heterogeneity, both between and within subpopulations of players, include using a model of computational errors and the allowing for diversity in prior beliefs around a modal prior for the subpopulation.
A related– and competing– theory of hierarchical bounded rationality is Cognitive hierarchy theory which assumes a more restrictive Poisson distribution.
- ^Stahl, D. O. (1993). Evolution of Smartn Games and Economic Behavior, 5(4), 604-617.
- ^Haruvy, D. Stahl, and P. Wilson. Evidence for optimistic and pessimistic behavior in normal-form games. Economics Letters, 63:255–259, 1999.
- ^ O. Stahl and E. Haruvy. Level-n bounded rationality and dominated strategies in normal-form games. Journal of Economic Behavior and Organization, 66(2):226–232, 2008a.
- ^Haruvy, E., & Stahl, D. O. (2012). Between-game rule learning in dissimilar symmetric normal-form games. Games and Economic Behavior, 74(1), 208-221
- ^Stahl II, D. O., & Wilson, P. W. (1994). Experimental evidence on players’ models of other players. Journal of Economic Behavior & Organization, 25(3), 309-327.
- ^Nagel, Rosemarie. “Unraveling in Guessing Games: An Experimental Study”. The American Economic Review, Vol. 85, Issue 5. December 1995
- ^Stahl, Dale and Wilson, Paul. “On Players’ Models of Other Players: Theory and Experimental Evidence”. Games and Economic Behavior. 10, 1995
- ^Haruvy, Ernan; Stahl, Dale; Wilson, Paul (2001). “Modeling and Testing for Heterogeneity in Observed Strategic Behavior”. Review of Economics and Statistics. 83: 144–157.
Ofer Abarbanel is a 25 year securities lending broker and expert who has advised many Israeli regulators, among them the Israel Tax Authority, with respect to stock loans, repurchase agreements and credit derivatives. Founder of TBIL.co STATX Fund.