Probabilistic Choice (Optional)

by choosing Right only 24% of the time in the 40 treatment, and 74% of the time in the 360 treatment.

The qualitative effect of Row’s own-payoff effect is captured by a noisy-response model where the sharp-cornered best noisy-response functions on the left sides of Figures 10.3 and 10.4 are rounded off, as shown on the right sides of the figures. Notice that the intersection of the curved lines implies that Row’s proportion of Top choices will be below 1/2 in the 40 treatment and above 1/2 in the 360 treatment, and the predicted change in the proportion of Right decisions will not be as extreme as the movement from 0.1 to 0.9 implied by the Nash prediction. The actual data averages are shown by the black dots on the right sides of Figures 10.3 and 10.4. These data averages exhibit the strong own-payoff effects that are predicted by the curved best-response lines, and the prediction is fairly accurate, especially for the 40 treatment.

The quantitative accuracy of these predictions is affected by the amount of curvature that is put into the noisy response functions, and is ultimately a matter of estimation. Standard estimation techniques are based on writing down a mathematical function with a “noise parameter” that determines the amount of curvature and then choosing the parameter that provides the best fit. The nature of such a function is discussed next.

Using Luce’s suggestion, the next step is to find an increasing function, i.e. one with a graph that “goes uphill.” (Mathematically speaking, a function, f , is increasing if a f(π1) > f(π2) whenever π1 > π2, but the intuitive idea is that the slope in a graph is from lower-left to upper-right.) Several examples of increasing functions are the linear function: f(π1) = π1, or the exponential function, f(π1) = exp(π1). The linear function obviously has an uphill slope; its graph is just the 45-degree line. The exponential function has a curved shape, like a hill that keeps getting steeper, like a snow-capped Mt. Fuji in a Japanese woodblock print.

Once we have found an increasing function, we might be tempted to assume that the probability of the decision is determined by the function itself:

Pr(D1) = f(π1) and Pr(D2) = f(π2). The problem with this approach is that it does not ensure that the two probabilities sum to 1. This is easily fixed by a simple trick with a fancy name: “normalization.” Just divide each function by the sum of the functions:

(10.1) ₁ ¹ ₂ ²

1 2 1 2

( ) ( )

Pr( ) and Pr( ) .

( ) ( ) ( ) ( )

f f

D D

f f f f

π π

π π π π

= =

+ +

If you are feeling a little unsure, try adding the two ratios in (10.1) to show that Pr(D1) + Pr(D2) = 1. Now let’s see what this gives us for the linear case:

(10.2) ₁ ¹ ₂ ²

1 2 1 2

Pr( )D π and Pr(D ) π .

π π π π

= =

+ +

Suppose π1 = π2 = 1. Then each of the probabilities in (10.2) will equal 1/(1+1) = 1/2. This result holds as long as the expected payoffs are equal, even if they are not both equal to 1. This makes sense; if each decision is equally profitable, then there is no reason to prefer one over the other, and the choice probabilities should be 1/2 each. Next notice that if π1 = 2 and π2 = 1, the probability of choosing decision D1 is higher (2/3), and this will be the case whenever π1 > π2.

One potential limitation to the usefulness of the payoff ratios in (10.2) is that expected payoffs may be negative if losses are possible. This problem can be avoided if we choose a function in (10.1) that cannot have negative values, i.e.

f(π1) > 0 even if π1 < 0. This non-negativity is characteristic of the exponential function, which can be used in (10.1) to obtain:

(10.3) ₁ ¹ ₂ ²

1 2 1 2

exp( ) exp( )

Pr( ) and Pr( ) .

exp( ) exp( ) exp( ) exp( )

D π D π

π π π π

= =

+ +

This avoids the possibility of negative probabilities, and all of the other useful properties of (10.2) are preserved. The probabilities in (10.3) will sum to one, they will be equal when the expected payoffs are equal, and the decision with the higher expected payoff will have a higher choice probability. The probabilistic choice model that is based on exponential functions is known as the logit model.

The curved lines in the right parts of the figures in this chapter were constructed using logit choice functions, but not quite the ones in equation (10.3).

It is true that (10.3) applied to the expected payoffs for the matching pennies games will produce curved response functions, but the lines will not have as much curvature as those in the figures in this chapter. The lines drawn with equation (10.3) have corners that are too sharp to explain the “own payoff effects” that we are seeing in the matching pennies games. Just as we could make it harder to distinguish between the width of two pins by making them each half as thick, we can add more noise or randomness into the choice probabilities in (10.3) by reducing all expected payoffs by a half or more. Intuitively speaking, dividing all expected payoffs by 100 may inject more randomness, since dollars become pennies, and non-monetary factors (boredom, indifference, playfulness) may have more influence. The right panels of the figures in this chapter were obtained by using the logit model with all payoffs being divided by 10:

(10.4) ₁ ¹ ₂ ²

1 2 1 2

exp( /10) exp( /10)

Pr( ) , Pr( ) .

exp( /10) exp( /10) exp( /10) exp( /10)

D π D π

π π π π

= =

+ +

At this point, you are probably wondering, why 10, why not 100? The degree to which payoffs are “diluted” by dividing by larger and larger numbers will determine the degree of curvature in the noisy response functions. Thus we can think of the number in the denominator of the expected payoff expressions as being an error parameter that determines the amount of randomness in the predicted behavior. The logit error parameter will be called µ, and it is used in the logit choice probabilities:

(10.5) ₁ ¹ ₂ ²

1 2 1 2

exp( / ) exp( / )

Pr( ) , Pr( ) .

exp( / ) exp( / ) exp( / ) exp( / )

D π µ D π µ

π µ π µ π µ π µ

= =

+ +

Extensions and Further Reading

The use of probabilistic choice functions in the analysis of games was pioneered by McKelvey and Palfrey (1995), and the intersections of noisy best response lines in Figures 10.2-10.4 correspond to the predictions of a quantal response equilibrium. (A “quantal response” is essentially the same thing as a noisy best response.) The approach taken in this chapter is based on Goeree, Holt, and Palfrey (2004, 2005), who introduce the idea of a regular quantal response equilibrium, and who discuss its empirical content and numerous applications. Goeree, Holt, and Palfrey (2000) use this approach to explain behavior patterns in a number of matching pennies games. Their analysis also includes the effects of risk aversion, which can explain the over-prediction of own-payoff effects for high payoffs that is seen on the right side of Figure 10.3.

Risk aversion introduces diminishing marginal utility that reduces the attractiveness of the large 360 payoff for the row player, and this shifts that person’s best response line down so that the intersection is closer to the data average point.

All of the games considered in this chapter involved two decisions, but the same principles can be applied to games with more decisions (see Goeree and Holt, 1999; Capra, Goeree, Gomez, and Holt, 1999, 2001, which will be discussed in the next two chapters). Of particular interest is Composti (2003), who had University of Virginia soccer players attempt penalty kicks into a net that was divided into three sectors by hanging strips of cloth. Thus the kicker had three decisions corresponding to the sector of the kick, say L, M, and R. Similarly, the goalie had three decisions, corresponding to the direction of the defensive dive.

In a symmetric treatment, the kicker received $3 for each point scored, regardless of sector, and the goalie received $3 each time a kick was unsuccessful for any reason. The penalty kicks were approximately evenly divided between the two sides, 13 attempts to L and 11 attempts to R, with only 6 to the middle. In an asymmetric treatment, the kickers received $6 for each kick scored on the R side, and $3 for each kick that scored to the M or L sides. The goalie’s incentives did not change ($3 for each kick that did not score, regardless of direction). This asymmetry in the kicker’s payoffs caused a strong own-payoff effect, with 25 attempts to R, 5 to L, and none to the middle. This study exhibits the advantages of an experiment, where asymmetries can be introduced in a manner that is not possible from an analysis of data from naturally occurring soccer games.

The focus in this chapter has been on games, but probabilistic choice functions can be applied to simple decision problems. For example, recall that a risk neutral person would make exactly 4 safe choices in the lottery choice menu presented in Chapter 4, and a person with “square root utility” would make 6 safe choices. These predictions produce lines with sharp corners, e.g. the dashed line in Figure 4.2, which looks a lot like the curved lines in the figures in this chapter.

The actual data averages shown in that figure have smoothed corners that would result from a probabilistic choice function, and Holt and Laury (2001) did find that such a function provided a good explanation of the actual pattern of choice proportions.

Questions

1. In a matching pennies game played with pennies, one person loses a penny and the other wins a penny, so the payoffs are 1 and –1. Show that there is a simple way to transform the game in Table 10.1 (with payoffs of 36 and 72) into this form. (Hint: first divide all payoffs by 36, and then subtract a constant from all payoffs. This would be equivalent to paying in 36-cent coins if they existed, and charging an entry fee to play.)

2. Consider a soccer penalty kick situation where the kicker is equally skillful at kicking to either side, but the goalie is better diving to one side.

In particular, the kicker will always score if the goalie dives away from the kick. If the goalie dives to the side of the kick, the kick is always blocked on the goalie’s right but is only blocked with probability one half on the goalie’s left. Represent this as a simple game, in which the goalie earns a payoff of +1 for each blocked kick and −1 for each goal, and the kicker earns −1 for each blocked kick and +1 for each goal. A 0.5 chance of either outcome results in an expected payoff of 0. Determine the equilibrium probabilities used in the Nash equilibrium.

3. Show that the two choice probabilities in (10.5) sum to 1.

4. Show that doubling all payoffs, i.e. both π1 and π2, has the same effect as reducing the noise parameter µ in (10.5) by half.

5. Show that multiplying all payoffs in (10.2) by 2 will not affect choice probabilities.

6. Show that adding a constant amount, say x, to all payoffs in (10.5) will have no effect on the choice probabilities. Hint: Use the fact that an exponential function of the sum is the product of the exponential functions of the two components of the product: exp(π+x) = exp(π)exp(x).