Overview

The question we will be addressing in this exercise is whether there is any evidence of loss/disappointment aversion in tournaments.

Background

This exercise can be regarded as a test between two paradigms of utility maximisation. Neoclassical economics postulates that utility – and therefore economic agents’ actions – depend only on final states. For example, you prefer to have 100 pounds over having only 90 pounds. Some behavioural economists claim that utility depends on final states relative to a reference point. In this exercise the reference point will be (approximately) the expected payoff. A disappointment averse agent might prefer (or behave as if he preferred) having 90 pounds when he was expecting to have only 50 over having 100 pounds when he was expecting to have 150.

Theory

Two players are competing for a prize (e.g. a promotion) of value B > 0. Only the winner gets B, the loser gets 0. The competition is sequential. First, player 1 chooses his effort level ex. Player 2 observes this effort before choosing his own effort level e. Here we will only be concerned with the second player’s chosen effort given the effort of the first player

(we will not model the choice of ex by player 1). Both players can choose an effort level from the set [0, e ̄]. The probability of winning for player 2 is given by

P = e − ex + γ 2γ

where we assume γ ≥ e ̄ (this ensures that P is between 0 and 1). The direct utility of player 2 is given by u(y) where y denotes the payoff (B or 0) and u(B) > u(0). Effort is costly and given by a convex cost function c(e).

1. Write down the expected utility function of player 2 as a function of e and ex.

2. Derive the optimal effort level e. Show that this effort does not depend on the first player’s effort ex.

3. Demonstrate the effect of the prize size B on effort.

4. Now suppose that instead of evaluating utility at the payoff level, player 2 evaluates his utility by comparing the payoff to a fixed reference payoff R. That is, the “effective” payoffs are B − R and 0 − R. Does effort depend on ex in this case?

5. Now let us make the following adjustments. The utility function of player 2 depends on whether he exceeds or falls short of a reference utility level R. We will define u as payoff utility, and U as experienced utility. The player maximises his (expected) experienced utility. If u(y) > R then experienced utility is

U(e, ex) = u(y) + G ∗ (u(y) − R) − c(e) and if u(y) ≤ R then experienced utility is

U(e, ex) = u(y) + L ∗ (u(y) − R) − c(e)

where G > 0 and L < 0. We are interested in “disappointment-averse” players, where disappointment aversity is defined as −L > G. This means that falling short of expectations by a given payoff utility level R−u(y) decreases your experienced utility MORE than exceeding your expectations by the same level u(y) − R increases it. Let us also simplify the payoff utility by assuming u(y) = y, and that the reference utility R is the expected payoff utility given the levels of effort e and ex:

R = E(y|e, ex) = B ∗ P (e, ex)

Write down the expected utility of player 2 in terms of P, G, L, B, and the cost function c(e). You don’t need to substitute the formula for P. Try to simplify the expression by using λ = L − G (disappointment aversion means that λ is positive).

6. Let the cost function be c(e) = qe + re2 . Solve the maximisation problem of player 2

2 and find the optimal effort of player 2 as a function of the effort of player 1. If we assume 2γ2r − Bλ > 0 (looks arbitrary, but is needed to make the expected utility function concave, so in line with economic conventions), does player 2’s effort react to an increase in player 1’s effort? Explain this result in an intuitive way.