The second‐tier trap: Theory and experimental evidence

• Author: Duk Gyoo Kim
• Published date: 01 December 2018
• DOI: http://doi.org/10.1111/ijet.12158
• Date: 01 December 2018

doi: 10.1111/ijet.12158

The second-tier trap: Theory and experimental evidence

Duk Gyoo Kim∗

Winner-take-all competitions can lead to the person in the second-tier (middle-tier) environ-

ment having the worst expected payoff when players exclusively choose their environment and

exert effort before their random, heterogeneous environmental supports are realized. The tiers

are defined by the ranks in pairwise competitions. The second-tier trap (STT) is a situation in

which a player from the second-tier environment has the worst expected payoff even though

his expected environmental support is strictly greater than that of the third-tier player. A suf-

ficient condition for the STT is that the ex-ante advantages, the winning probabilities when all

the players exert the same amount of effort regardless of their environment, are the same for

those two environments. I claim that this sufficient condition for the STT is so weak that players

can easily be tempted to choose the second-tier environment, which is the wrong decision. Lab

experiments strongly support this claim.

Key wor ds contests, all-pay auction, Lazear–Rosen model, laboratory experiment

JEL classification C72, C91, D81

Accepted 6 April 2017

1 Introduction

I consider a winner-take-all competition among nplayers in which their “environmental supports”

are random and heterogeneous. In the first phase, playersexclusively choose an environment to which

they want to belong, based on some informative statistics about the environments. In the second

phase, they choose their effort level, and after that their environmental support is realized. The

second phase of the game can be interpreted in a manner similar to that of Lazear and Rosen (1981),

namely, that the environmental support is the random or luck component. It is well known that in a

many-player competition with a winner-take-all payoffstructure, the variance of the environmental

support (or of some random component in other contexts) can significantly affect the players’

strategies and payoffs in equilibrium, and therefore it is straightforward to predict that the ranking

of the expected environmental supports may not be consistent with the ranking of the expected

utilities of the players.

My main research question was what ex-ante information is sufficient for choosing an optimal

(i.e., expected-payoff-maximizing) environment, and whether economic agents will indeed choose

the optimal environment. I was particularly interested in the situation where the best environment

∗Division of the Humanities and Social Sciences, California Institute of Technology, Pasadena, CA, USA. Email:

dgkim@caltech.edu

I would especially like to thank an anonymous referee, Robert H. Frank, and Stephen Coate. I also thank JohnAbowd,

Marco Battaglini, Larry Blume, Ted O’Donoghue, PradeepDube y, Alex Imas, Jun Sung Kim, Sang-HyunKim, William

D. Schulze, and Rasmus Wiesefor their helpful comments. I gratefully acknowledge financial support from the Cornell

Population Center and WilliamD. Schulze. All remaining errors are mine.

International Journal of Economic Theory 14 (2018) 323–349 © IAET 323

International Journal of Economic Theory

The second-tier trap Duk Gyoo Kim

is not allowed to be chosen, so agents are asked to pick the second-best environment. The second-

tier trap (STT) is a situation in which a player from the second-tier environment (the one with

the second-largest expected environmental support) has the worst expected payoff even though his

expected environmental support is strictly greater than that of the third-tier player.That is, when the

best environment is unavailable and a sufficient condition for the STT is observed, players should

choose the third-tier environment. I call this situation the STT because players are easily tempted to

choose the second-tier environment over the second-best environment. The sufficient condition for

the STT is so weak that even a sophisticated player could mistake the second-tier environment for

the second-best one.

Though the model has the form of a competition among many identical players who exclusively

choose one environment each, it would also serve as counterfactual analysis of an individual’s irre-

versible life choices. We often encounter a situation where we must choose one of several exclusive

options that will affect our life for a substantial period of time thereafter. Which college should I go

to? Which major? Which career? Whichsocial group? Even more challengingly, in several situations

where someone else has chosen the most preferable option, or we are not acceptedfor that option, we

have to choose one of the remainingoptions. The situation where the best choice is taken by someone

else does not necessarily imply a difference in the abilities of those individuals (Frank 2016). Identi-

cal n-player rank-order tournaments and Tullock contests theoretically predict the following: every

player exerts the same amount of effort, and one of them is randomly chosen to be the winner.In this

case, we cannot say the winner is better than the others or that the winner exerts more effort. Indeed,

as the number of competitors gets larger, it is more likely that the first-best option has alreadybeen

taken by someone else, and so the other players haveto choose the best option still available to them.

This paper provides a novel approach toaddressing this issue in the form of a two-stage competition.

Just like all life choices, a decision-maker chooses an exclusive environment, and chooses an effort

level to compete with other “counterfactual selves” who chose the road not taken. Since considering

many selves, instead of many identical players, does not change the model and its predictions, all

“identical players” in this paper may be thought of “counterfactual selves.” However, for the sake of

expositional simplicity, consistency with the laboratory experiment design, and potential relation to

the existing literature, I maintain the interpretation of the model as a many-player competition.

In the first stage of my model, identical players reveal their preferences ofenvironments and are

assigned to environments according those preferences. If two or moreplayers have the same prefer-

ence, they are randomly assigned to different environments with equal probability.Alternatively, this

could be thought of as a random ordering of identical players who take turns choosing the environ-

ment to which they want to belong. The player’s environmentalsupport is r andomly drawn froman

environment-specific support distribution at the end of the game. The support distribution can be

interpreted as a different market situation that each player faces;1a characteristic of the group,such

as a team, school, career,or social identity (Akerlof and Kranton 2005); or simply a distribution of the

luck component. Players know the support distributions, and are able to calculate some informative

statistics, such as means, variances, and ex-ante advantages, at the time they are given a choice of

environments. The ex-ante advantage of an environment is defined as the winning probability of a

1As an example of many-player competition, considera situation w hereone of three workers will be promoted based on

their outputs (the sum of their effort and the growth rate of the market), and the three workers will be in charge of

separate international markets whose potential growth rates are random and heterogeneous. Before choosingtheir effor t

level, they have to exclusively chooseone mar ketwhose g rowthr ateis not yet realized. As an example of counterfactual

analysis, consider a situation where a high school senior chooses a major for college. He tries to maximize his well-being

(expected payoff minus cost of effort) but does not know which major will enable him to get a job that pays the most

four years later.

324 International Journal of Economic Theory 14 (2018) 323–349 © IAET