The conditions A to F for social and collective player is’ dilemmas show asymmetries between player j and players i.
Consider the case where 𝑥1= 𝑥2= 𝑒𝑗. From (4), if 𝑒𝑗 = 0, the marginal private return for each player i is −1 + (1
3)𝑀𝐿. The marginal private return for player j from (5) if 𝑥1= 𝑥2= 0 is -1. Comparing (4) and (5) when 𝑥1 = 𝑥2= 𝑒𝑗 = 0, the marginal private return is higher from (4); (1
3) 𝑀𝐿 > 0. This indicates that the marginal private return for a player i is higher than the marginal private return for the player j when 𝑒𝑗 = 0.
Consider now that 𝑥1= 𝑥2= 1 or both players i contribute fully to the PGG. The marginal private return for a player i from (4) is now −1 + (1
3)[𝑀𝐿+ (1 −)(𝑀𝐻− 𝑀𝐿)]. For player j, the marginal private return from (5) when 𝑥1= 𝑥2= 1 is −1 + (1
3)(2)(1 −)(𝑀𝐻− 𝑀𝐿). Comparing the marginal private returns of (4) and (5) when there are maximum contributions and effort by others, (1
3) [𝑀𝐿+ (1 −)(𝑀𝐻)] > 0, that player i’s marginal private return on contribution is higher than player j’s when 𝑒𝑗= 1.
The marginal social return for a player i is 3 ∗ [−1 + (1
3) [𝑀𝐿+ (1 − )(𝑀𝐻− 𝑀𝐿)]] > 3. This will result in (1
6) [𝑀𝐿+ (1 − )(𝑀𝐻− 𝑀𝐿)] > 0. The marginal social
18 return for player j is 3 ∗ [−1 + (1
3) (2)(1 − )(𝑀𝐻− 𝑀𝐿)] > 3 and this results in
1
3∗ (1 − )(𝑀𝐻− 𝑀𝐿)>0. Substituting the values of the values of 𝑀𝐻, 𝑀𝐿, and would yield a bigger marginal social return for a player i compared to the player j.
Figure 2.1 gives a graphical representation of the possible returns to players i and j relative to the Nash-equilibrium strategy benchmark in which their respective payoff equals 1, π =1, at zero contribution and zero effort by all players. To interpret Figure 2.1, we use the expected multiplier value that player j’s effort produces and derive payoffs for the group and everyone from these values. The probability element in multiplier determination will be incorporated in the experiment’s implementation in Sarawak.
The space of the figure shows all possible combinations of effort by player j and average contributions by players i. This space is divided into regions according to the benefits received by the two types of players. A player’s benefit is defined as the actual payoff they receive minus the Nash-equilibrium payoff of 1. The vertical axis refers to the effort space available to player j while the average contribution of players i is shown on the horizontal axis, i.e.( 𝑥1+ 𝑥2)/2. When discussing the figure, we will assume that both players i are contributing equally given their contribution space.
Figure 2.1. Possible benefits from public good based on combinations of effort and average contribution to public good
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The dotted 45-degree line in Figure 2.1 represents the equality between, i) average payoff to the two players i, i.e. ( 𝜋𝑖1+ 𝜋𝑖2)/2, and ii) the payoff to player j, i.e. 𝜋𝑗. The origin point of the graph is the non-interaction or Nash-equilibrium benchmark, in which both effort and contribution is zero. The solid black curve shows the combination of effort and contributions that result in 𝜋𝑗 = 1, i.e player j’s payoff is equal to what she/he would have got in Nash-equilibrium. Player j’s payoff from maximum effort, 𝜋𝑗 is 1 when average contribution, ( 𝑥1+ 𝑥2)/2, is 0.706. It is the same payoff she/he would have made if she/he made zero effort and both players i had made zero contributions. On the other hand, the grey line repre sents the combination of decisions by both types of players that result in 𝜋𝑖 = 1, i.e. the average payoff to players i is equal to to what they would have got in Nash equilibrium. When player j’s effort is 0.285, an average player i’s payoff 𝜋𝑖 is 1, independent of their average contributions.
The point where the grey line and solid black curve cross marks the point at which positive effort and positive contributions produce an average payoff of the two players i, ( 𝜋𝑖1+ 𝜋𝑖2)/2 and a payoff of player j, 𝜋𝑗, both of which are equal to 1, i.e.
the payoff from exerting effort and contributing is equal to the payoff of zero effort and zero contribution. Here, player j must allocate 28.5% of her/his endowment as effort and both players i must contribute 28.5% of her/his endowment to the public good for everyone to receive a payoff that equals the Nash -equilibrium benchmark payoff.
Regions I, II, III and IV are defined by the solid black curve and grey horizontal line.
If combinations of effort and average contribution fall in Region I, the payoffs for all players are less than 1. Region I is identified as a region of mutual losses, indicating that everyone is better off not engaging with the public good. For example, the average player i contribution is 0.1 and the effort from player j is 0.1. Combinations of decisions in Region II benefit an average player i at a cost to player j. An average player i receives a payoff more than 1, 𝜋𝑖1+𝜋𝑖2
2 > 1 , while player j’ payoff is less than 1, 𝜋𝑗 < 1. In this region, player j is engaging in leading-by-sacrifice or is acting prosocially. Region III produces a mutually beneficial outcome for everyone and encompasses the area below the solid black curve and above the horizontal grey line.
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On the other hand, combinations of decisions in Region IV benefit player j, 𝜋𝑗 > 1 but at a cost for an average player i, 𝜋𝑖1+𝜋𝑖2
2 < 1.
Decision in Region III is characterised with combinations of effort and contributions benefitting both types of players beyond the Nash-equilibrium benchmark value. At the northeast point of the 45-degree dotted line is the socially optimal point.
Within Region III combinations of decisions could also result in one player type receiving bigger benefits than the other type. If decisions end up in between the 45-degree line and the horizontal grey line, the share of benefit is larger for player j compared to the average player i. On the other hand, share of benefits from the PGG is larger for the average player i than for player j if the combination of effort and contributions settles somewhere in between the solid black curve and the 45 -degree line.
If the representative can be trusted to play her/his part by exerting effort at any point above the grey horizontal line, players i collectively will receive a positive return from their contributions. This covers Regions II and III. On the other hand, a representative benefits from effort as long as her/his effort corresponds with an average contribution value that is on the right side of the solid black curve. This refers to Regions III and IV. If her/his effort falls in Region III, the representative has played her/his role in the game and enhanced benefits for the group members. If her/his effort falls in Region IV, then the representative is free riding on the collective action by the group members. In the situation where players i and j match each others’
decisions, their combination of decisions will end up on the 45 -degree line and everyone will receive equal payoff from the public good, 𝜋𝑗= 𝜋𝑖1= 𝜋𝑖2.