Optimal strategies and cost-benefit analysis of the $${varvec{n}}$$ n -player weightlifting game

Preliminaries

To unify all the five classes of two-by-two games, Yamamoto et al.³⁵ introduced the weightlifting game. In this game, each player either cooperates or defects in carrying a weight. Players who carry the weight pay a cost, (cge 0). The weight is successfully lifted with probability ({p}_{i}), where (i=mathrm{0,1},2) is the total number of cooperators and ({p}_{i}) increases with the number of cooperators (i). If the cooperators succeed, both players receive a benefit (b>0). However, in case of failure, both players gain nothing. The pay-off of the cooperators is (b{p}_{i}-c), and the pay-off of the defectors is (b{p}_{i}) (Table 2). In terms of the parameters (Delta {p}_{1}={p}_{1}-{p}_{0}) and (Delta {p}_{2}={p}_{2}-{p}_{1}), which represents the increase in the probability of success due to an additional cooperator, the following inequalities are obtained for the pay-offs (R, T, S), and (P) (Table 1):

1. (i)

   (Delta {p}_{1}>c/b) for (S>P),

2. (ii)

   (Delta {p}_{2}>c/b) for (R>T), and

3. (iii)

   (Delta {p}_{1}+Delta {p}_{2}>c/b) for (R>P).

PD satisfies only (iii), CH satisfies (i) and (iii), SH satisfies (ii) and (iii), DT satisfies none of the three conditions, and CT satisfies all three. In 2021, Chiba et al.¹ studied the evolution of cooperation in society by incorporating environmental value in the weightlifting game. They found that the evolution of cooperation seems to follow a DT to DT trajectory, which can explain the rise and fall of human societies.

The ({varvec{n}})-player weightlifting game

In this study, we generalize the weightlifting game to (n)-players. Suppose (n) self-interested and rational individuals selected from a population of infinite size. The (n) players are asked to lift a weight. Each individual (or player) can decide to either carry the weight (cooperate, (C)) or not carry/pretend to carry the weight (defect, (D)). Players who decide to carry the weight can either succeed or fail. The probability of successful weightlifting is denoted by ({p}_{i}), (i=mathrm{0,1},dots ,n), where (i) indicates the number of cooperators (henceforth, (i) always represents the number of cooperators). The probability of success increases with the number of individuals cooperating, and it may remain less than unity even if all (n) individuals cooperate. Players who decide to carry the weight pay a cost, (cge 0), regardless of the outcome, while those who defect need not pay anything. If the cooperators succeed, all (n) individuals receive a benefit (bge 0). There is no penalty for failure. We use the expected gains/losses of the players as the pay-off. If there are (i-1) cooperative players, then the pay-off of (j) is ({B}_{C}left(iright)=b{p}_{i}-c) when (j) cooperates and ({B}_{D}left(i-1right)=b{p}_{i-1}) when (j) defects. The number of cooperators differs by one, since in ({B}_{C}left(iright)), there is an additional cooperator, which is (j) him- or herself. To decide whether to cooperate or defect, all players weigh their expected gain and rationally choose the option with the highest expected gain. The graphical outline of this game is illustrated in Fig. 1 (see also Supplementary Figure S1 for the flow of the game). The pay-off table for a four-player game is shown as an example in Table 3. Here, player (1) is the innermost row (strategies are listed in the second column of the table), player (2) is the innermost column (strategies are listed in the second row of the table), and the succeeding players take the succeeding rows or columns (we enter the first player as a row player and the following player as a column player and continue in this order). Each cell represents players’ pay-offs, with the first component being the pay-off for the first player, the second for the second player, and so on. For instance, consider the entry in the first row and third column, where players (1, 2) and (3) cooperate but player (4) defects. The pay-offs of players (1) to (3) are ({B}_{C}(3)), while the pay-off of player (4) is ({B}_{D}left(3right)). In the above example, there are as many row players as column players because the number of players is even. However, we can have one more player in the rows than in the columns if there is an odd number of players.

Nash equilibrium and pareto optimal strategies

Here we present the Nash equilibrium and Pareto optimal strategies of the (n)-player weightlifting game in terms of the cost-to-benefit ratio (c/b) and probability of success ({p}_{i}). The Nash equilibrium consists of the best responses of each player. Players have no incentive to deviate from this strategy profile since deviation will not increase an individual’s pay-off if the other players maintain the same strategy. If ({B}_{C}(i)ge {B}_{D}(i-1)), the best response of player (j) is to cooperate, but if ({B}_{C}(i)le {B}_{D}(i-1)), the best response is to defect.

We have (Delta {p}_{i}={p}_{i}-{p}_{i-1}ge 0) for the increase in the probability of success because the probability ({p}_{i}) increases with the number of cooperators (i). It is convenient to divide cases depending on whether (Delta {p}_{i}>c/b) or (Delta {p}_{i}<c/b). We obtain the following results (see Supplementary Text for the derivations):

Result 1 If (Delta {p}_{1}le c/b), there is a Nash equilibrium at (left(D,D,dots ,Dright)). The Nash equilibrium at ((D,D,dots ,D)) is unique if and only if (Delta {p}_{i}<c/b), for all (i=mathrm{1,2},dots ,n).

Result 2 If (Delta {p}_{n}ge c/b), there is a Nash equilibrium at (left(C,C,dots ,Cright)). The Nash equilibrium at ((C,C,dots ,C)) is unique if and only if (Delta {p}_{i}>c/b), for all (i=mathrm{1,2},dots ,n).

Result 3 There is a Nash equilibrium in the combination of strategies where (i-1) players choose (C) and the rest of the players choose (D) if and only if (Delta {p}_{i}<c/b<Delta {p}_{i-1}), for some (i=mathrm{2,3},dots ,n).

Result 1 shows that players have no incentive to cooperate when the cost relative to the benefit is (very) high, so much so that (Delta {p}_{i}<c/b), for all possible values of (i). This case of all defection is a unique equilibrium, where no player can improve the pay-off by cooperating. In contrast, Result 2 shows that all players cooperate when the cost is sufficiently smaller than the benefit. Results 1 and 2 indicate that cooperation is determined by the relationship between the cost and the benefit; raising the benefit or lowering the cost can increase cooperation. There may be cases where full defection or cooperation is not a unique equilibrium (see cases 3 or 10, for example, in Table 4). The reason for this is covered by Result 3. This result shows the conditions for the existence of equilibria where only some individuals cooperate, which we will refer to as anti-coordination equilibria. Result 3 also implies the significance of an individual in promoting cooperation. For instance, when (Delta {p}_{2}<c/b<Delta {p}_{1}), we have an equilibrium with a single cooperator. While there is a small chance of success, if an individual’s contribution to the probability of success is substantial, cooperation will exist. These three results cover all possible cases of pure equilibrium. The equilibria at (left(D,D,dots ,Dright)) and at (left(C,C,dots ,Cright)) are covered by Results 1 and 2, respectively, and the anti-coordination equilibria are covered by Result 3.

Result 4 The number of equilibria of an (n)-player weightlifting game is at most (sumnolimits_{i = 0}^{{leftlfloor frac{n}{2} rightrfloor }} {Cleft( {n,2i} right)}) if (n) is even and (sumnolimits_{i = 0}^{{leftlfloor frac{n}{2} rightrfloor }} {Cleft( {n,2i} right) + 1}) if (n) is odd, where (Cleft( {n,2i} right)) denotes the combination of (2i) out of (n).

Result 4, on the other hand, gives the maximum number of equilibria in a weightlifting game. To illustrate this result, the equilibrium strategies (marked with X) of a four-player game are presented in Table 4. Notably, the one X in case 2 means not just one equilibrium but four equilibria: ((C,D,D,D)), (left(D,C,D,Dright),) (left(D,D,C,Dright)) and (left(D,D,D,Cright)). The same applies to the other cases (except 1 and 16). As shown in Table 4, there can be at most three types of equilibrium (case 11): all-(D), anti-coordination, and all-(C). There is exactly one all-(D) and exactly one all-(C) strategy. However, there are (left(2+2right)!/(2!2!)=C(mathrm{4,2})) anti-coordination equilibria of two players cooperating and two players defecting; thus, there are at most eight equilibria in a four-player game. This finding is in accordance with Result 4: ({sum }_{i=0}^{2}C(mathrm{4,2}i)=Cleft(mathrm{4,0}right)+Cleft(mathrm{4,2}right)+Cleft(mathrm{4,4}right)=8).

In Pareto optimal strategies, players cannot increase their pay-offs by changing their strategy without also decreasing the other players’ pay-offs. Owing to ({p}_{i}le {p}_{i+1}), ({B}_{D}left(iright)le {B}_{D}left(i+1right)) and ({B}_{C}left(iright)le {B}_{C}(i+1)). Thus, if a defector cooperates, the rest of the players will enjoy an increased pay-off. Moreover, some players will suffer from a decreased pay-off if cooperators decrease. In this case, we only have to check the condition that makes a strategy profile Pareto-dominated, i.e., when defectors cooperate.

Result 5 Strategy (left(C,C,dots ,Cright)) is Pareto optimal if and only if ({sum }_{j=1}^{n}Delta {p}_{j}>c/b).

Result 6 The strategy profile with (i) defectors, (i=mathrm{0,1},dots ,n-1), is Pareto optimal if and only if ({sum }_{j=i+1}^{n}Delta {p}_{j}<c/b.)

In ((C,C,dots ,C)), the only way a player can deviate is to defect; thus, it is sufficient to check the condition where all-(D) Pareto-dominates all-(C). However, in the following result, which covers the remaining strategies, all-(D) does not Pareto-dominate these strategies since defectors are disadvantaged. Furthermore, we know that ({sum }_{j=1}^{n}Delta {p}_{j}) saturates towards unity. Thus, intuitively, cooperation is Pareto optimal unless (c) is close to or greater than (b.)

General properties of the ({varvec{n}})-player games

While Yamamoto et al.³⁵ considered only the conditions that encourage cooperation, the violation of these conditions implicitly implies the satisfaction of the converse conditions. Thus, PD, SH and DT satisfying (Delta {p}_{1}<c/b) assures equilibrium at ((D,D)). Moreover, PD and DT satisfying (Delta {p}_{2}<c/b) makes this equilibrium unique, according to Result 1. On the other hand, SH satisfying (Delta {p}_{2}>c/b) leads to another equilibrium at ((C,C)) (Result 2). The anti-coordination equilibrium of CH is covered by the condition (Delta {p}_{2}<c/b<Delta {p}_{1}) of Result 3. In addition, the condition (Delta {p}_{1}+Delta {p}_{2}>c/b) (condition iii), which PD, CH, SH and CT satisfy, indicates that all-(C) is more beneficial than all-(D). As in Result 5, the counterpart of this condition for the (n)-player game is ({sum }_{j=1}^{n}Delta {p}_{j}>c/b). Similarly, the inequality ({sum }_{j=1}^{n}Delta {p}_{j}<c/b) indicates that all-(D) is more beneficial than all-(C) in Result 6 when (i=0).

The five classes of two-by-two games are characterized by their equilibria and optimal strategies. All these games are unified under a single structure in the two-player weightlifting game. As an extension of the (n)-player game, the following correspondence occurs: With all-(C) being the optimal strategy, PD has a unique equilibrium at all-(D), CH has an anti-coordination equilibrium, SH has an equilibrium at both all-(C) and all-(D), CT has a unique equilibrium at all-(C), and DT has a unique and optimal equilibrium at all-(D). In the above results, we have shown the existence and uniqueness of an equilibrium and the existence of optimal strategies. In summary, we present the conditions and characterization of the (n)-player games in Table 5.

Illustration

Let us consider a concrete example of lifting a weight of (W=100) by four individuals (Figs. 2 and 3). The weight that each individual cay carry is normally distributed with mean (mu) and standard deviation (sigma). For (mu =10) and (sigma =50), we obtain (Delta {p}_{1}=0.013,Delta {p}_{2}=0.019,Delta {p}_{3}=0.026) and (Delta {p}_{4}=0.034) (Figs. 2a1, 2a2). The n-player CT obtains for (0<c/b<0.013), SH for (0.013<c/b<0.034), PD for (0.034<c/b<0.092), and DT for (0.092<c/b<1). In Figs. 2a1 and 2a2, we show the parameter regions for Nash equilibria and Pareto optimal strategies as hatched in the i–(c/b) plane, where i is the number of cooperators and (c/b) is the cost-to-benefit ratio. As (c/b) increases, the number (i) of cooperators drops from four to zero in Nash equilibria (Fig. 2a1). In Pareto optimal strategies, the number (i) decreases from four to zero, while the range of (c/b) for (i=0sim 3) reaches the right end point (c/b=1) (Fig. 2a2). Note that the boundary values for the hatched bars are different for Nash equilibria (Fig. 2a1) and Pareto optimal strategies (Fig. 2a2). Similarly, we obtain Figs. 2b–e and 3a–e for (mu) from 20 to 100. The range for each game category varies depending on (mu). As (mu) increases, SH ceases to exist (Fig. 2e) while the coexistence CH&PD begins to appear (Fig. 2d) and disappear (Fig. 3e). A pure CH appears afterwards (Fig. 3b).

Source: Ecology - nature.com