Level-k reasoning models finite chains of strategic best responses. A beauty contest shows how the level-0 anchor, opponent mixture and own impact change the answer.
Level k reasoning
Level-k reasoning is a model of strategic choice in which players form beliefs by taking a finite number of best-response steps from a specified level-0 starting point. In the basic version, level 1 responds to level 0, level 2 responds to level 1, and so on. For a quant modelling an auction or a competitor's trading decision, it offers an explicit alternative to assuming that everyone already anticipates everyone else's strategy correctly.
The best-response idea comes from game theory: choose an action that maximises expected payoff against a stated belief about others. A higher level changes that belief. It does not automatically make the belief correct. Costa-Gomes and Crawford (2006), pp. 1738–1739, distinguish these response rules from equilibrium and from rules based on iterated dominance.
A ladder anchored at 50
Consider a toy beauty contest. Each participant chooses a real number between 0 and 100. The target is p=2/3 times the population average, and each minimises the squared distance between their number and the target. Treat the population as sufficiently large that one person's choice does not move that average.
This quadratic-loss teaching version makes the best response transparent. Nagel's original experiment instead rewarded the contestant closest to the target with a prize; the payoff rules should not be confused.
Let M denote the average choice the player expects to face. Under squared loss,
E[(x−pM)2]=(x−pE[M])2+p2Var(M).
The expected value of the average determines the best response: x=pE[M]. The variance term affects expected loss but does not move its minimiser. Assume a level-0 mean of 50—for example, the mean of uniform choices on [0,100].
Model level
Belief about opponents
Chosen number
0
Non-strategic starting mean
50
1
Opponents have mean 50
100/3≈33.33
2
Opponents use level 1
200/9≈22.22
3
Opponents use level 2
400/27≈14.81
Thus xk=50(2/3)k. These are thought experiments inside one decision, not successive trading rounds or observations of actual people. The number 50 is an assumption that anchors the model; it is not a theorem about an uninformed population. Nagel's §II motivates this anchor and finite iteration. Level 0 can also function only as the higher levels' model of others, without requiring an actual population of level-0 players; see Crawford, Costa-Gomes and Iriberri, §2.4.
Toy quadratic-loss beauty contest with p equal to two-thirds and own impact ignored: the level-0 mean is 50; pure level-1, level-2 and level-3 choices are 33.33, 22.22 and 14.81. A level-2 response to equal perceived weights on levels 0 and 1 is 27.78 instead of 22.22.
Original calculations for stipulated beliefs, not experimental observations. The ladder uses pure lower-level beliefs; the lower panel changes level 2 to a specified mixture of lower levels.
Two steps against whom?
Cognitive hierarchy changes the basic belief rule: a level-k player responds to a mixture of levels below k, rather than only level k−1. Camerer, Ho and Chong (2004), §II.A, specify lower-level weights using a population distribution, renormalised over the levels the player considers possible.
For an original comparison, stipulate that a level-2 player puts equal weight on level 0 and level 1. Keeping the same large-population squared-loss setup, the perceived mean is
21(50)+21(3100)=3125.
The response becomes (2/3)(125/3)=250/9≈27.78, rather than 22.22. These weights describe the player's belief; they are not measured population shares. Saying only “two levels of reasoning” leaves the action undetermined until the opponent model is specified.
Put your own choice back into the average
For a small group, the earlier approximation can materially change the answer. Suppose there are n players and the mean of the other n−1 choices is m. Your own number x makes the overall mean (x+(n−1)m)/n. With the same quadratic loss, the zero-error choice solves
x=pnx+(n−1)m⟹x=n−pp(n−1)m.
With three players, p=2/3 and other-player mean m=50, this is 200/7≈28.57, compared with the negligible-impact answer 33.33. The number is feasible in [0,100], and the coefficient on x in the error is nonzero, so it is the unique squared-loss minimiser. If m is uncertain, replace it by its expected value. The finite-group correction tends to pm as n grows. Nagel flags own influence in footnotes 7–8; the calculation here applies it explicitly to our stipulated loss.
Depth is a model parameter, not a trading advantage
A Nash equilibrium requires mutual best responses. If everyone chooses zero in this toy game, choosing zero is optimal against them. The finite level-k choices above instead respond to beliefs that need not match actual play. Although their ladder tends to zero as k grows, this example does not establish convergence of level-k reasoning in other games.
More steps can even worsen the forecast of a shallow population. If the actual mean is 50 and own impact is negligible, level 1 hits the target exactly. Level 2 chooses 22.22 and misses the target of 33.33. The extra step helps only if it improves the opponent belief relevant to the payoff.
For empirical use, specify the game, level-0 rule, response errors and candidate types before interpreting fitted levels. One action need not identify a person's depth: an anchor of 75 with two steps gives the same 100/3 as an anchor of 50 with one step. Costa-Gomes and Crawford's introduction and §II.D explain why variation across games and tests for omitted decision rules matter. For a market application, evidence that a laboratory type fits a participant does not establish that the same rule predicts their quotes under inventory limits, private information or learning. A useful model must survive those changes and predict decisions beyond the observations used to fit it.
Sources
Rosemarie Nagel, Unraveling in Guessing Games: An Experimental Study, American Economic Review 85(5), 1313–1326 (1995), opening game description, §II and footnotes 7–8: finite reasoning, the 50 anchor and the own-impact caveat.
Colin F. Camerer, Teck-Hua Ho and Juin-Kuan Chong, A Cognitive Hierarchy Model of Games, Quarterly Journal of Economics 119(3), 861–898 (2004), §II.A: responses to a renormalised mixture of lower levels.
