Effort-Cost Effects

Next consider what happens when the cost of effort is altered. For example, suppose that the effort cost of 10 used to construct Table 12.1 is raised to 19, so that the payoffs (for an effort of E and a minimum effort of M) will be determined by: 60 + 20M - 19E. In this case, a one unit increase in each person’s effort raises payoffs by 20 minus the cost of 19, so the payoffs in the upper-right box of Table 12.3 are only 1 cent higher than the payoffs in the lower-left box.

Notice that this increase in effort cost did not change the fact that there are two Nash equilibria, and that both players prefer the high-effort equilibrium.

However, simple intuition suggests that effort levels in this game may be affected by effort costs. From the row player’s perspective, the top row offers a possible gain of only one cent and a possible loss of 19 cents, as compared with the bottom row.

Table 12.3. A Minimum-Effort Game With High Effort Cost (Row’s payoff, Column’s payoff)

Column Player:

Row Player: Low Effort (1) High Effort (2) High Effort (2) 42, 61 62, 62

Low Effort (1) 61, 61 61, 42

Goeree and Holt (1999, 2001) report experiments in which the cost of effort is varied between treatments, using the payoff formula:

(12.2) own payoff = M −cE,

where M is the minimum of the efforts, E is one’s own effort, and c is a cost parameter that is varied between treatments. Effort decisions were restricted to be any amount between (and including) $1.10 and $1.70. As before, any common effort is a Nash equilibrium in this game, as long as the cost parameter, c, is between 0 and 1. This is because a unilateral decrease in effort by one unit will reduce the minimum by 1, but it will only reduce the cost by an amount that is less than 1. Therefore, a unit decrease in effort will reduce one’s payoff by 1 – c.

Conversely, a unilateral increase in effort by one unit above some common level will not raise the minimum, but the payoff will fall by c. Even though deviations from a common effort are unprofitable when c is greater than 0 and less than 1, the magnitude of c determines the relative cost of “errors” in either direction. A large value of c, say 0.9, makes increases in effort more costly, and a small value of c makes decreases more costly.

Figure 12.1 shows the results for sessions that consisted of 10-12 subjects who were randomly paired for a series of 10 rounds. There were three sessions with a low effort cost parameter of 0.25, and the averages by round for these sessions are plotted as thin dashed lines. The thick dashed line is the average over all three sessions of this treatment. Similarly, the thin solid lines plot round-by-round averages for the three sessions with a high effort cost parameter of 0.75, and the thick solid line shows the average for this treatment.

Efforts in the first round average in the range from $1.35 to $1.50, with no separation between treatments. Such separation arises after several rounds, and average efforts in the final round are $1.60 for the low-cost treatment, versus

$1.25 for the high-cost treatment. Thus we see a strong cost effect, even though any common effort is a Nash equilibrium.

One session in each treatment seemed to approach the boundary, which raises the issue of whether behavior will “lock” on one of the extremes. This pattern was observed in a Veconlab classroom experiment in which efforts went to $1.70 by the 10^th period. This kind of extreme behavior is not universal, however. Goeree and Holt (1999) report a pair of sessions that were run for 20 rounds. With an effort cost of 25 cents, the decisions converged to about $1.55, and with an effort cost of 75 cents the decisions leveled off at about $1.38. Both of these outcomes seem to fit the pattern seen in Figure 12.1.

The spreadsheet-based analysis of noisy behavior for the Travelers’

observed in the experiments, even though these data patterns are not predicted on the base of an analysis of the Nash equilibria for the game.

$1.00

$1.10

$1.20

$1.30

$1.40

$1.50

$1.60

$1.70

$1.80

0 1 2 3 4 5 6 7 8 9 10

Round

Average Effort

low-cost sessions c = 0.25

high-cost sessions c = 0.75

Figure 12.1. Average Efforts for the Goeree and Holt (1999) Coordination Experiment:

Thick Lines are Averages Over Three Sessions for Each Cost Treatment

IV. Extensions

Goeree and Holt (1999) also considered the effects of raising the number of players per group from 2 to 3, holding effort cost constant, which resulted in a sharp reduction of effort levels. Effort-cost effects were observed as well in games in which payoffs were determined by the median of the three efforts.

Some coordination games may be played only once; Goeree and Holt (2001) report strong effort-cost effects in such games as well. A theoretical analysis of these effects (in the context of a logit equilibrium) can be found in Anderson, Goeree, and Holt (2001b).

There is an interesting literature on factors that facilitate coordination on good outcomes in matrix games, e.g. Sefton (1999), Straub (1995), and Ochs (1995). In particular, notions of “risk dominance” and “potential” provide good predictions about behavior when there are multiple equilibria. Anderson, Goeree, and Holt (2001b) introduce the more general notion of stochastic potential and use it to explain results of coordination game experiments. Models of learning

and evolution have also been widely used in the study of coordination games (see references in Ochs, 1995).

Appendix: An Analysis of Noisy Behavior in the Coordination Game

The traveler’s dilemma and the coordination game have similar payoff structures, so it is straightforward to modify the spreadsheet in Table 11.2 so that it applies to the coordination game. Here are the steps:

Step 1. Save the traveler’s dilemma spreadsheet under a different name, e.g.

cg.xls, before deleting all information in column L and farther to the right.

Step 2. The coordination experiment has fewer possible decisions, when restricted to 10-cent increments: 110, 120, … 170. Therefore, delete all material in rows 7-10, and in rows 18-20, leaving row 21 as it is. You will have to enter the possible effort levels in column A: 110 in A11, 120 in A12, etc.

Step 3. There is no penalty/reward parameter in the coordination game, so enter a 0 in cell B2. Next put “C =” in cell A3, and enter a value of 0.75 in cell B3.

Step 4. The starting values for the cumulative column sums will have to be moved, so put values of 0 in cells F10 and H10.

Step 5. To set the initial probabilities, change the entry in cell D11 to 1, and leave the other entries in that column to be 0.

Step 6. Next we have to subtract the effort cost from the expected payoff formula in cell I11. The cost per unit effort in cell B3 must be multiplied by the effort in column A to calculate the total effort cost. Thus you should append the term

−$B$3*$A11 to the end of the formula that is already in cell I11. Then this modified formula should be copied into rows 12 to 17 in this column.

Step 7. At this point, the numbers you have should be essentially the same as those in Table 12.4 (which have been truncated), with a new vector of probabilities in column K. To perform the first iteration, mark the block from E10 to K21, copy, and place the cursor in cell L10 before pasting. This same

Table 12.4. Excel Spreadsheet for Coordination Game Logit Responses

A B C D E F G H I J K

1 µ = 10 2 R = 0 3 C = 0.75 4

5

6 X X X P PX F(X) PX π^e Exp(π^e/µ) P

7 8 9

10 100 0 0

11 110 110 110 1 110 1 110 110 27.5 15.64 0.530 12 120 120 120 0 0 1 0 110 20 7.38 0.250 13 130 130 130 0 0 1 0 110 12.5 3.49 0.118

14 140 140 140 0 0 1 0 110 5 1.64 0.056

15 150 150 150 0 0 1 0 110 -2.5 0.77 0.026 16 160 160 160 0 0 1 0 110 -10 0.36 0.012 17 170 170 170 0 0 1 0 110 -17.5 0.17 0.006 18

19 20

21 1 ¹¹⁰ 429.491 1

The expected effort after one iteration, found in L21, is about 119. This rises to 125 by the fifth iteration. When the initial probabilities in column D are changed to put a probability of 1 for the highest effort and 0 for all other efforts, the iterations converge to about 125 by the 10^th iteration. When the cost of effort is reduced from 0.75 to 0.25, the average claims converge to about 155 after several iterations. These average efforts are quite close to the levels for the two treatment averages in Figure 12.1 in round 10. As noted in the previous chapter, the convergence of iterated stochastic responses indicates an equilibrium in which a given set of beliefs about others’ effort levels will produce a matching distribution of effort choice probabilities. The resulting effort averages are quite good predictors of behavior in the minimum-effort experiment when the error rate is set to 10. Even though any common effort is a Nash equilibrium, intuition suggests that lower effort costs may produce higher efforts. These spreadsheet calculations of the logit equilibrium are consistent with this intuition; they explain why efforts are inversely correlated with effort costs. A lower error rate will

push effort levels to one or another of the extremes, i.e. to 110 or to 170 (question 3).

Questions

1. Consider the game in the table below, and find all Nash equilibria in pure strategies. Is this a coordination game?

Column Player:

Row Player: Left

(26%) Middle

(8%) Non-Nash

(68%) Right (0%) Top

(68%)

200, 50 0, 45 10, 30 20, −250 Bottom

(32%) 0, −250 10, −100 30, 30 50, 40

2. The numbers under each decision label in the above table show the percentage of people who chose that strategy when the game was played only once (Goeree and Holt, 2001). Why is the column player’s most commonly used strategy labeled as “Non-Nash” in the table? (Note: it was not given this label in the experiment.) Conjecture why the Non-Nash decision is selected so frequently by column players?

3. Use the spreadsheet for the minimum-effort coordination game to fill in average efforts for the following Table. Use initial belief probabilities of 1/8 (or 0.125) for each of the elements in column D between rows 11 to 17. Then discuss the effects of the error parameter on these predictions.

Average Effort Predictions

µ = 1 µ = 5 µ = 10 µ = 20 High Cost

(c = 0.75)

Low Cost

magnitude of X affects the frequency with which the Row Player chooses Top, so the issue is whether this effect is predicted in theory. Find all Nash equilibria in pure and mixed strategies for X = 0, and for X = 400, under the assumption that each person is risk neutral.

Column Player:

Row Player: L R S

Top 90, 90 0, 0 X, 40 Bottom 0, 0 180, 180 0 , 40

5. (Advanced) Find the mixed-strategy equilibrium for the game shown in question 1. (Hint: In this equilibrium, column randomizes between Left and Middle. The probabilities associated with Non-Nash and Right are zero, so consider the truncated 2x2 game involving only Left and Middle for column and Top and Bottom for row. Finally, check to be sure that column is not tempted to deviate to either of the two strategies not used.)