Game Theory Explains Perplexing Evolutionary Stable Strategy Math 188: Social Choice and Decision Making

Erin N. Bodine April 15, 2003

Abstract This paper reviews a letter published in Nature by researchers B. Sinervo and C.M. Lively on “The rock-paper-scissors game and the evolution of alternative male strategies.” This paper explains what Sinervo and Lively studied, and how they modeled what they observed. Additionally, comments are given on how this model differs from a typical evolutionary stable strategy (ESS) model and a few suggestions on how the model could be extended in future research are discussed.

The Biology Male side-blotched lizards (Uta stansburiana) come in three morphs or types: aggressive, less aggressive and ‘sneakers’. Each type of male has a different coloring on their throat and defends a different size territory. The aggressive males have orange throats (see Figure 1 left) and defend a large territory. The less aggressive males have dark blue throats (see Figure 1 middle) and defend a smaller territory than the aggressive type males. The sneaker males have yellow stripes across their throats (see Figure 1 right) and defend no territory. The sneaker males get their nickname from looking like females receptive for mating who also have yellow striping across their throats. In “The rock-paper-scissors game and the

Figure 1: The various physiological forms of the male Uta stansburiana. The left images show the orange throated or aggressive male Uta stansburiana. The middle images show the dark blue throated or less aggressive male Uta stansburiana. The right images show the yellow throated or ‘sneaker’ Uta stansburiana. evolution of alternative male strategies” Sinervo and Lively are interested in explaining the territory 1

use and patterns of sexual selection of the male Uta stansburiana. They collect data over a six year period (1990-95) keeping track of the frequency of each type of Uta stansburiana male. Sinervo and Lively find that over the six years each Uta stansburiana male type has a period of dominance (i.e. it is the male type of the highest frequency). The periods of dominance are shown in Table 1. Sinervo Male Type blue/less aggressive orange/aggressive yellow/‘sneaker’

Years of Highest Frequency 1991, 1995 1992 1993-94

Table 1: Years of highest frequency for each of the male types of Uta stansburiana. The blue or less aggressive type had the highest frequency in 1991 and 1995, the orange type in 1992, and the yellow type in 1993-94. and Lively then calculate the fitness of each type of male for each year. The fitness of each morph of male is estimated to be the number of females within the male’s territory (monopolized females) plus the male’s share of females in a shared territory (shared females). The fitness values of each morph of male for 1991, 1992 and 1994 are given in Table 2.

Year 1991 1992 1994

Orange 4.73 1.28 0.90

Fitness Blue 1.00 1.00 1.26

Yellow 1.61 1.45 1.00

Table 2: Fitness of each morph of Uta stansburiana for the years 1991, 1992, and 1994. The cell of the table containing the value 4.73 is read as the orange or aggressive male type had a fitness value of 4.73 in 1991. Typically, intra-species population models follow an evolutionary stable strategy (ESS) model; that is, one morph always has better fitness regardless of its frequency. However, viewing the frequencies from Table 1 and the fitness values from Table 2 it is easily seen that this is not the case. There exists no one morph of male Uta stansburiana that always has the highest fitness value. In 1991, the aggressive (orange) male has the highest fitness, but in 1992, the ‘sneaker’ (yellow) male has the highest fitness. In 1994, the type of male with the highest fitness changes again, this time to the less aggressive (blue) male. Since an ESS model clearly will not work, how is the intra-species population of Uta stansburiana males to be modeled? Given this perplexing situation, Sinervo and Lively turn to the field of game theory and the classic game of rock-paper-scissors to explain the typical mating situation of the Uta stansburiana.

The Model To model the biological phenomena they find, Sinervo and Lively compute a payoff matrix. A payoff matrix is a matrix where each entry in the matrix is the relative payoff between two competing strategies. The payoffs in Sinervo and Lively’s payoff matrix are “frequency dependent ‘fitness payoffs’ of a rare morph competing against a common morph from the ‘local’ frequency-dependent interaction among males” [1], where the rare morph is the male type with the lowest frequency at a given time. The

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payoff matrix Wij computes the fitness payoff for a rare ith strategy against a common j th strategy. The matrix is computed as follows Wij = [ Mi + Ni × GMij ] + [ Si + Ni × GSij ] where Mi is the fitness acquired through monopoly of female territories, Si is the fitness acquired through shared females, Ni is the number males that the ith strategy must compete against, and G is the slope of the line describing the gains and losses for the ith strategy for each neighbor of the j th strategy [1]. Using this method of computing, graphs of relative fitness of a rare morph vs. a common morph and of morph frequency can be generated (see Figure 2). Any of the three strategies would be a

Figure 2: Graphs showing relative fitness and morph frequencies. Figure a shows the relative fitness of rare morphs versus common morphs. Figure b shows the morph frequency of the Uta stansburiana projected over 100 generations. ESS if it maintains a higher fitness no matter which morph of male Uta stansburiana is rare. However, Figure 2 shows that this is not the case. When each morph is held up and compared to the other morphs when they are rare, each morph only maintains a higher fitness than one of the other morphs. Thus, Figure 2a shows that each morph has an advantage over one morph and has no advantage or 3

a disadvantage against the other. When the yellow or sneaker male is the common morph it has an advantage against the blue males, but a disadvantage against the orange males. When the dark blue or less aggressive male is the common morph it has an advantage against the orange males, but has no advantage against the yellow males. When the orange or aggressive male is the common morph it has an advantage against the yellow males, but a disadvantage against the blue males. This circular set of advantages is displayed in Figure 3. In the concluding paragraph of their letter to Nature Sinervo

Figure 3: In this diagram Sinervo and Lively draw the connection between relative fitness of male Uta stansburiana and the game of rock-paper-scissors. Here the orange or aggressive male Uta stansburiana is the rock, the yellow or ’sneaker’ male Uta stansburiana is the paper, and the dark blue or less aggressive male Uta stansburiana is the scissors. Corresponding to the rock-paper-scissors game this image shows that orange loses to yellow, yellow loses to dark blue, and dark blue loses to orange. and Lively draw the connection between the relative fitness of male Uta stansburiana in mating and the game of rock-paper-scissors. In this closing paragraph they state, “We have described the first biological example of a cyclical ‘Rock-paper-scissors’ game” [1]. Though Sinervo and Lively do not explain how, exactly, the rock-paper-scissors model is applicable, with a little investigation it becomes clear. In order to gain a clear understanding of why Sinervo and Lively’s model is like a rock-paperscissors game model, consider how game theorists typically model the rock-paper-scissors game. First, rock-paper-scissors is game between two players. The payoffs of this game for player I are described by the game matrix in Table 3. Thus, if two Uta stansburiana lizards are the players and their strategies are blue, yellow and orange, or aggressive, less aggressive and sneaker, then the outcome of the game will resemble the rock-paper-scissors game (though the payoffs are not exactly -1, 0, and 1). Table 3 shows the payoff matrix for the Uta stansburiana mating fitness game. This game is similar to the rock-paper-scissors

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The Game Matrix for Rock-Paper-Scissors r s p

R 0 -1 1

S 1 0 -1

P -1 1 0

Table 3: Game matrix for the game of rock-paper-scissors. Player I has the choice between the strategies r, s, and p which correspond to rock, scissors, and paper, respectively. Player II has the choice between the strategies R, S, and P which correspond to rock, scissors, and paper, respectively. Within the matrix a value of -1 translate into a loss for player I (or a win for player II), 0 into a draw between player I and player II, and 1 into a win for player I. The Game Matrix for Uta stansburiana Strategies

Common Morph

b y o

Rare Morph B Y O 1.0 2.0 0.2 1.0 1.0 2.0 4.5 0.4 1.0

center around zero

→

Common Morph

b y o

Rare Morph B Y O 0.0 1.0 -0.8 0.0 0.0 1.0 3.5 -0.6 0.0

Table 4: Game matrix for Uta stansburiana mating strategies. The rare morph will either be a blue (less aggressive), yellow (sneaker), or orange (aggressive) type lizard. This corresponds with the B, Y , and O strategies. The common morph will also either be a blue (less aggressive), yellow (sneaker), or orange (aggressive) type lizard. This corresponds with the b, y, and o strategies. The game on the left corresponds with the values found in Figure 2a. The game on the right has had 1 subtracted from the value of each payoff in the game on the left. games in that no one strategy dominates (i.e. has an advantage) over all other strategies. One might suggest that this model does not make sense because the lizards do not get to “choose” their strategy. However, instead of viewing this as two specific lizards which get to choose their strategy, it can be thought of as nature randomly putting two Uta stansburiana lizards in the same territory, each lizard then representing one player. Of course, this all becomes more complicated as morph frequencies are considered, but the underlying model for the fitness of male Uta stansburiana in mating is still the rock-paper-scissors model since each strategy has an advantage against one other strategy and a disadvantage against the other.

Possible Extensions and Improvements to the Model One logical extension to the model Sinervo and Lively present, would be to attempt to model the fitness of male Uta stansburiana in mating as a three player game with each male type as one player. It would then be interesting to compare the results from this model with the results of the Sinervo and Lively model. It would also be interesting to extend the model to include any the Uta stansburiana’s natural predators. Do the Uta stansburiana have any natural predators? If so, do they prey on the certain male morphs of Uta stansburiana more than others? If so, how would this effect the intra-species population dynamics? 5

Lastly, it would be interesting to collect more data to see if the six year cycle holds for more than one period.

Conclusions Game theory is useful for modeling many types of situations. Researchers Sinervo and Lively use game theory to explain an evolutionary stable intra-species population dynamic among three male morphs of Uta stansburiana. Prior to the study of Uta stansburianas, evolutionary stable models consisted of a series of strategies where one strategy was more fit than the rest. The example of the side-blotch lizards shows that there are other sets of strategies which produce evolutionary stable population dynamics.

References [1] Sinervo, B. and C.M. Lively. 1996. The rock-paper-scissors game and the evolution of alternative male strategies. Nature 388: 240 - 243. [2] Straffin, P.D. 1993. Game Theory and Strategy. Mathematical Association of America, Washington DC.

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