The Effects of Payoff Imbalances

As one would expect, the fact that the noisy best response lines always intersect in the center of Figure 10.2 is due to the balanced nature of the payoffs for this game (Table 10.1). An unbalanced payoff structure is shown in Table 10.2, where the Row player’s payoff of 72 in the Top/Left box has been increased to 360. Recall that the game was balanced before this change, and the choice proportions should be 1/2 for each player. The increase in Row’s Top/Left payoff from 72 to 360 would make Top a more attractive choice for a wide range of beliefs, i.e. Row will choose Top unless Column is almost sure to choose Right.

Intuitively, one would expect that this change would move Row’s choice proportion for Top up from the 1/2 level that is observed in the balanced game.

This intuition is apparent in the choice data for an experiment done with Veconlab software, where the payoffs were in pennies. Each of the three sessions involved 10-12 players, with 25 periods of random matching. The proportion of Top choices was 67% for the “360 treatment” game in Table 10.2.

Table 10.2. An Asymmetric Matching Pennies Game (Row’s payoff, Column’s payoff)

Column Player:

Row Player:

Left Right Top 360, 36 ⇒ ⇓ 36, 72

Bottom 36, 72 ⇑ ⇐ 72, 36

This intuitive “own payoff effect” of increasing Row’s Top/Left payoff is not consistent with the Nash equilibrium prediction. First, notice that there is no equilibrium in non-random strategies, as can be seen from the arrows in Table 10.2, which go in a clockwise circle. To derive the mixed equilibrium prediction,

let p denote Row’s beliefs about the probability of Right, so 1–p is the probability of Left. Thus Row’s expected payoffs are:

Row’s Expected Payoff for Top = 360(1–p) + 36(p) = 360 –324p.

Row’s Expected Payoff for Bottom = 36(1–p) + 72(p) = 36 + 36p.

It follows that the difference in these expected payoffs is: (360 – 324p) – (36 + 36p), or equivalently, 324 – 360p. Row is indifferent if the expected payoff difference is zero, i.e. if p = 324/360 = 0.9. Therefore, Row’s best response line stays at the top of the left panel in Figure 10.3 as long as the probability of Right is less than 0.9. The striking thing about this figure is that Column’s best response line has not changed from the symmetric case; it rises from the Bottom/Left corner and crosses over when the probability of Top is 1/2. This is because Column’s payoffs are exactly reflected (36 and 72 on the Left side, 72 and 36 on the right side). In other words, the only way that Column would be willing to randomize is if Row chooses Top and Bottom with equal probability.

In other words, the Nash equilibrium for the asymmetric game in Table 10.2 requires that Top and Bottom be played with exactly the same probabilities (1/2 each) as was the case for the case of balanced payoffs in Table 10.1. The reason is that Column’s payoffs are the same in both games, and Row must essentially use a coin flip in order to keep Column indifferent.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Probability of Top

360 data

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Probability of Top

The previous treatment change made Top more attractive for Row by raising Row’s Top/Left payoff from 72 to 360. In order to produce an imbalance that makes Top less attractive, we reduce Row’s Top/Left payoff in Table 10.1 from 72 to 40. Thus Row’s expected payoff for Top is: 40(1–p) + 36(p) = 40 – 4p, and the expected payoff for Bottom is: 36(1–p) + 72(p) = 36 + 36p. These are equal when p = 4/40, or 0.1. What has happened in Figure 10.4 is that the reduction in Row’s Top/Left payoff has pushed Row’s best response line to the bottom of the figure, unless Column is expected to play Right with a probability that is less than 0.1. This downward shift in Row’s best response line is shown on the left side of Figure 10.4. The result is that the best response lines intersect where the probability of Top is 1/2, which was the same prediction obtained from the left sides of Figures 10.2 and 10.3. Thus a change in Row’s Top/Left payoff from 40 to 72 to 360 does not change the Nash equilibrium prediction for the probability of Top. The mathematical reason for this result is that Column’s payoffs do not change, and Row must choose each decision with equal probability or Column would not want to choose randomly.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Probability of Right

Probability of Top

40 data

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Probability of Right

Probability of Top

Figure 10.4. The 40 Treatment:

Best Responses (Left Side) and Noisy Best Responses (Right Side)

These invariance predictions are not borne out in the data from the Veconlab experiments reported here. Each of the three sessions involved 25 periods for each treatment, with the order of treatments being alternated. The percentage of Top choices increased from 36% to 67% when the Row’s Top/Left payoff was increased from 40 to 360. The Column players reacted to this change

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.