Periodic frequencies of the cycles in $2\times 2$ games: evidence from experimental economics

The original aim of Setlen and Chumra’s experimental investigation is to compare the performance of the five stationary concepts selten2008 . They designed twelve $2\times 2$ cyclic matching-pennies-like games Traulsen2005 ; cremer2008anomalous . Games 1 to 6 are constant-sum games, and each of them contains 12 groups; games 7 to 12 are non-constant-sum games, and each of them contains 6 groups. In addition, games 7 to 12 have the same Nash equilibria as the games 1 to 6, respectively.

Each of the 108 groups consists of two four-person populations, so a total of 864 subjects are involved in the experiments. Each group plays the game for 200 rounds repeatedly. The strategy set for each player in the first population (denoted by $X$) is ($X_{1}$,$X_{2}$); while the strategy set for each player in the second population (denoted by $Y$) is ($Y_{1}$,$Y_{2}$). In each round, every subject plays the game against an opponent randomly picked from the other population. The payoff matrix for each game has the same form as Fig. 1 shows, where $a_{ij}$ ($b_{ij}$) indicates the payoff for the player in the first (second) population, $i,j=1,2$. The specific value of $a_{ij}$ ($b_{ij}$) in each game is exhibited in Table 1.

As an approximation, evolutionary game dynamics can be a powerful tool to describe finite population games Borgers1997 ; Friedman1998 ; Samuelson2002 ; Friedman2012 . It takes the social state, i.e. the combination of the densities of strategies in each population Sandholm2011 , as a crucial object and analyses its motion in state space. In this case, each social state is denoted as $x$:=$(p,q)$, where $p$ ($q$) is the density of $X_{1}$ ($Y_{1}$) in the first (second) population $X(Y)$ and the state space is $\mathbb{X}$=$\{0,\frac{1}{4},\frac{2}{4},\frac{3}{4},1\}\times\{0,\frac{1}{4},\frac{2}{4},\frac{3}{4},1\}$, as in Fig. 2 (for more details, see Appendix A). According to evolutionary game dynamics, the social state will change in the state space.

As predictors, the two models—standard replicator dynamics and adjusted replicator dynamics—are used as references. The advantage of the two models is that they are parameter free and the most widely used Traulsen2005 ; Maynard1982evolution ; Sandholm2011 ; Friedman1996 ; cremer2008anomalous . The solutions in a (sufficiently small) neighbourhood of the Nash equilibrium can be calculated for the two dynamics for the twelve games. According to the calculation, the motions in these games are cyclic (see Appendix A) and equivalent to two-dimensional simple harmonic motions (see Appendix J in Maynard1982evolution ). Therefore, based on these two models, the observed social state is supposed to keep moving in the state space Taylor1978 ; Maynard1982evolution ; Weibull1997 ; Traulsen2005 and thus form endless cycles, which leads to the following theoretical predictions:

1. 1.

   The cycles, for all 12 games, exist and persist. Meanwhile, the directions of the cycles are all clockwise.

2. 2.

   The frequencies of the cycles are different (for numerical results see Table 1, in which $f_{s}$ and $f_{a}$ respectively indicate the predictions from standard and adjusted replicator dynamics).

3. 3.

   The radius of the cycles, for all 12 games, should remain unchanged over time.

With 12 experimental $2\times 2$ games, we verify the existence and persistence of the cycles; and test the frequencies of cycles quantitatively. Furthermore, we find that the observed periodic frequencies of the cycles are different among games. Moreover, the experimental frequencies are linearly, positively and significantly related to the theoretical frequencies.

To verify the cycles quantitatively, we look to Poincare section (tripwire) Friedman2012 ; Xu20134997 for counting cycles. First, we set a segment $S$ called the Poincare section Friedman2012 , as shown in Fig. 2, from $P_{c}=(\alpha,\beta)$ to $P_{e}=(\alpha,0)$, and we take $(\alpha,\beta)$ equal to the Nash equilibrium point ($x_{N},y_{N}$) (see Table 1) of each game. A transit is a directional segment from state $(x_{1},y_{1}$) observed at time $t$ to state $(x_{2},y_{2})$ at time $t+1$. These two segments could intersect at X as

According to Ref Xu20134997 , the value of $C_{i}$ for a transit from time $t$ to $t+1$ should be:

1. 1.

   $C_{i}=0$, if $X\not\in(P_{c},P_{e}]\cup x_{2}=x_{1}$

2. 2.

   $C_{i}=1$, if $X\in(P_{c},P_{e}]\cap x_{2}>\alpha>x_{1}$

3. 3.

   $C_{i}=-1$, if $X\in(P_{c},P_{e}]\cap x_{2}<\alpha<x_{1}$

4. 4.

   $C_{i}=\frac{1}{2}$, if $X\in(P_{c},P_{e}]\cap x_{2}>x_{1}\cap(x_{1}=\alpha\cup x_{2}=\alpha)$

5. 5.

   $C_{i}=-\frac{1}{2}$, if $X\in(P_{c},P_{e}]\cap x_{2}<x_{1}\cap(x_{1}=\alpha\cup x_{2}=\alpha)$

The accumulated counting number $C_{t_{0}},_{t_{1}}$ of the evolutionary trajectory during the time interval [$t_{0},t_{1}$] is defined ( see Eq.2 in Xu20134997 ) as

The accumulated counting number $C$ quantifies the net number of cycles around the reference point ($\alpha,\beta$). As mentioned Xu20134997 , such a quantity can help us to detect deterministic behaviours in a stochastic process. $C$ indicates the existence of cycles when it does not equal 0. Additionally, $C>0$ ($C<0$) indicates that the direction of the cycles is counterclockwise (clockwise). We use the subscript $(t_{0},t_{1})$ to specify the observation in time interval $[t_{0},t_{1}]$, like $C_{t_{0},t_{1}}$. Clearly, $C$ is time odd ($C_{t_{0},t_{1}}$ = - $C_{t_{1},t_{0}}$). By calculating $C$ and investigating how it changes with time, we can verify the existence, persistence and direction of the cycles.

In this article, the experimental periodic frequency ($f_{exp}$) is defined as the time averaged net cycles counted in an experimental trajectory, which referring to Xu20134997 , is

where $\sum_{i}C_{i}$ is the accumulated counting number of cycles in time interval [$t_{0},t_{1}$] (the time unit in these 12 experiments is discrete in the experimental round), and $\bar{f}_{exp}$ is the average ${f}_{exp}$ over 12 groups of each of games 1 to 6 or 6 groups of each of games 7 to 12.

Whenever a transit-$i$ crosses the Poincare section, the Euclidean distance (denoted as $x_{i}$) between the crossing dot and the centre $N$ can be measured. Furthermore, for a given time interval $[t_{0},t_{1}]$, the average radius of the cycles¹¹1This observable, as a vector, can be understood as the moment of force or torque in classical mechanics in physics. can be defined as

where $i$ (=$1,2,3,\cdots$) denotes the $i$-th time that the trajectory passes the Poincare section. The sum covers all of the transits across the Poincare section. For example, according to Eq. 4, the radius of the net cycles ($R_{1,5}$) in Fig. 2 is -0.115. We use this measurement to evaluate the change in radius with time, which is necessary for cycle testing Nowak2012 ; Friedman2012 .

Result 1: Cycles exist and persist, and the directions of net cycles in the “stochastic” trajectories meet the predictions of both evolutionary dynamics.

To test the existence of net cycles as in Xu20134997 , we calculate the average value $\bar{C}_{1,200}$ for the 12 games as well as the standard error of the mean (s.e.m) over the groups. The results are shown in Table 2. Using the Wilcoxon signed-rank test, we find that the $\bar{C}_{1,200}$ of all games are significantly less than 0, indicating the existence of cycles in all games except game 9,²²2However, two points should be mentioned. First, each of games 7 to 12 only has 6 groups while each of games 1 to 6 has 12 groups; second, theoretically, the frequencies of games 1, 7, 3 and 9 are small, indicating that the formation of each cycle needs more time. So we use half of the observations of games 1 to 6, again, $p$-value$>5\%$ in games 1, 2, 3 and 4. So we still use this empirical number of cycles when comparing with the theoretical value later. see Table 2. Using the 12 games as samples, we find that the existence of the net cycles is significant ($p$=$0.002$, Wilcoxon signed-rank test with $C_{1,200}=0$).

The Wilcoxon signed-rank test verifies the persistence of cycles by comparing $C_{1,100}$ and $C_{101,200}$ of the 12 games. Results show that the null hypothesis: $C_{1,100}$=$C_{101,200}$ cannot be rejected ($p$=0.695, two-tailed, $d.f.$=11), which indicates the persistence of cycles.

Since all $\bar{C}_{1,200}$ of the 12 games are negative (see Table 2), the empirical directions of the cycles are clockwise statistically, consistent with the directions predicted by both of the two dynamics models as shown in Fig. 4.

Result 2: The periodic frequency of the net cycles can be tested in data.

Before calculating the frequencies of cycles as in Xu20134997 , the number of cycles in the trajectory is counted. Fig. 3 presents the average number of the net cycles ($C_{1,t}$, see Eq. 2 for its explanation) along time $t$ ($t$ is up to 200 rounds in experiments, $t=200$) of each game. The 12 curves respectively represent the average cycles counted for the 12 experiments. For example, the red curve (Game 6) ends at about -10 at $t=200$. This indicates that in the 200-round game, about 10 cycles are counted on average over the 12 groups; meanwhile, the negative value implies the clockwise direction of the cycles.

The frequencies $f_{exp}$ of the 12 experiments are calculated according to Eq. 3. The results are shown in Table 2. The observed periodic frequencies of the 12 games are significantly different, which is consistent with the patterns of Fig. 3. Specifically, Eq. 3 indicates that the frequencies can be represented by the slopes of the $C_{1,200}$ curves in Fig. 3. Since the slopes are different for the 12 games, the frequencies are different.

In Section 5, we will show that the empirical frequencies of the cycles is an appropriate criterion for evaluating the models quantitatively.

Result 3: The radius of the cycles does not change with time in the experiments, which supports the predictions of both of the two models.

For the 200-round experiments, our hypothesis is that the radius of the first 100 rounds ($R_{1,100}$) is equal to that of the second 100 rounds ($R_{101,200}$). The calculated results of $R_{1,100}$ and $R_{101,200}$ of the 12 games are shown in Table 2. Using the groups in each game as samples, the radius does not change in each of the 12 games ($p$-values are exhibited in Table 2, last column), and therefore the equal hypothesis cannot be significantly rejected. Using the 12 games as samples, we find that the equal hypothesis cannot be significantly rejected ($p=0.308$, Wilcoxon signed-rank test). Therefore, we can say that the statement that the radius does not change with respect to time is statistically supported.