Social Dilemmas in Nature-Based Tourism Depend on Social Value Orientations

Nature-based tourism (NBT) is vulnerable to a rapid increase in visitors because natural resources are often open access. Market failure caused by over-exploitation of natural resources is an example of social dilemmas in common-pool resource systems. Game theory, which describes people’s decision making under conflicts, has been applied to the analysis of social dilemmas in NBT. However, previous studies use non-cooperative games assuming individualistic players and discuss the emergence of social dilemmas only in a limited situation. Here, we demonstrate, by developing a two-player non-cooperative game of wildlife viewing, that the traditional game-theoretic approach fails to find social dilemmas. By analysing the competition between tour operators (players) with different social value orientations (SVOs), we found that concentration of tours becomes a Pareto-inefficient Nash equilibrium (PINE) when both players are competitive. Whether the wildlife-viewing market is a Prisoner’s dilemma depends on players’ SVOs. Furthermore, we found that fair punishment on competitive players promotes rather than suppresses the emergence of PINE. Our results suggest that the diversity of SVOs is an essential factor in understanding social dilemmas in NBT.

Here, we identify the conditions for which the SPBM has Nash equilibria.

(F, F)-Nash
The strategy profile (F, F) is a Nash equilibrium if The individualistic, competitive, and prosocial players have positive values of α i . Hence, we can reduce the equilibrium condition to U TF 1 ≤ 0. Solving this inequality with respect to the initial encounter probability gives The condition for (F, F)-Nash is independent of players' SVOs.

(T, T)-Nash
The strategy profile (T, T) is a Nash equilibrium if where BR s 1 s 2 i denotes the condition for which player i chooses the best response s i to the other player's strategy s −i . If player i is individualistic (α i = 1 and β i = 0), If player i is competitive (α i = 1/ √ 2 and β i = −1/ √ 2), If player i is prosocial (α i = 1/ √ 2 and β i = 1/ √ 2), . (S7)

(T, F)-Nash and (F, T)-Nash
The strategy profile (T, F) is a Nash equilibrium if The first condition can be reduced to P ≥ P 0 , which is independent of players' SVOs. The equilibrium condition depends only on the SVO of player 2. If player 2 is individualistic, the condition for (T, F)-Nash is If player 2 is competitive, If player 2 is prosocial, . (S11) The condition for (F, T)-Nash is obtained in the same way.

Result
Tables S1-S3 list the conditions for Nash equilibria in the cases of δ < C/B, δ = C/B, and δ > C/B, respectively. G τ 1 τ 2 denotes a WVG with player i of type τ i ∈ {I, C, P}. The symbols I, P, and C are abbreviations for individualistic, competitive, and prosocial types, respectively. ∅ denotes the empty set. The proposition P ∈ ∅ means that the equilibrium does not exist. Table S1: Conditions for which the SPBM has Nash equilibria (δ < C/B)

Game (F, F)-Nash (F, T)-Nash (T, F)-Nash (T, T)-Nash
2 Pareto-efficient solutions of the OPBM As shown by equation (2), the OPBM of the WVG is Here, we identify the conditions for which the OPBM has Pareto-efficient solutions.

(F, F)-Pareto
The strategy profile (F, F) is not a Pareto-efficient solution if The condition for (F, F)-Pareto is The condition for (T, T)-Pareto is

(T, F)-Pareto and (F, T)-Pareto
The strategy profile (T, F) is not a Pareto-efficient solution if The condition for (T, F)-Pareto is As the OPBM is symmetric, (F, T)-Pareto always coexists with (T, F)-Pareto. Table S4 lists the conditions for which the OPBM has Pareto-efficient solutions. As the individual, competitive, and prosocial players have the same OPBM, the solution conditions are independent of players' SVOs.

Conditions for PINE
From the conditions for Nash equilibria and Pareto-efficient solutions (Tables S1-S4), we obtain  the conditions for PINE. Tables S5-S6 show the results in the cases of δ < C/B and δ ≥ C/B, respectively.

Punishment effect on Nash equilibria
Here, we impose punishment on the concentration of tours and examine its effect on the emergence of Nash equilibria. As shown by equation (11), the SPBM with punishment is

(F, F)-Nash
The punishment does not change the conditions for (F, F)-Nash.

(T, T)-Nash
The SPBM has (T, T)-Nash if When player i is individualistic, where 0 = /(B (1 − δ)). Since 0 > 0, we have When player i is competitive, which is equal to the best response condition in the original game without punishment (equation (S6)). When player i is prosocial,

(T, F)-Nash and (F, T)-Nash
The SPBM has (T, F)-Nash if The equilibrium condition depends only on the SVO of player 2. If player 2 is individualistic, If player 2 is competitive, If player 2 is prosocial, The condition for (F, T)-Nash is obtained in the same way.  Figure S1 shows the distribution of Nash equilibria over the parameter space D = {(δ, P ) | δ ∈ (0, 1), P ∈ [0, 1]}. The parameter values are the same as Figure 3 except for . The punishment does not influence the equilibrium distribution of G CC , while it decreases the region of (T, T)-Nash in all the other games.

Punishment effect on Pareto-efficient solutions
Here, we examine the effect of fair punishment on the emergence of Pareto-efficient solutions. As shown by equation (10), the OPBM with punishment is

(T, T)-Pareto
The strategy profile (T, T) is not a Pareto-efficient solution if The condition for (T, T)-Pareto is

(T, F)-Pareto and (F, T)-Pareto
The strategy profile (T, F) is not a Pareto-efficient solution if (u FF ∈ P(u TF )) ∨ (u FT ∈ P(u TF )) ∨ (u TF ∈ P(u TF )) ∨ (u TT ∈ P(u TF )) The condition for (T, F)-Pareto is As the OPBM is symmetric, (F, T)-Pareto always coexists with (T, F)-Pareto. Figure S2 shows the distribution of Pareto-efficient solutions over the parameter space D. The parameter values are the same as Figure 4 except for . The region of (T, T)-Pareto is narrower than in the game without punishment.

Examples of SVO-based non-cooperative games
Here, we apply the SVO-based approach to three non-cooperative games: the Hawk-Dove game, Battle of the Sexes, and Public Goods game (Nowak, 2006;Maschler et al., 2013). For simplicity, we assume that two players have the same SVO.

Hawk-Dove game
The OPBM of the Hawk-Dove game is expressed as where strategies H and D indicate Hawk and Dove, respectively. It is assumed that 0 < b < c.
If players are individualistic, the SPBM of the game is equal to the OPBM. The SPBM has two Nash equilibria (D, H) and (H, D). The objective payoff allocations given by the solutions are Pareto efficient.
If players are prosocial, the SPBM of the game is which has three Nash equilibria (D, H), (H, D), and (D, D). The objective payoff allocations given by the solutions are Pareto efficient. If players are competitive, the SPBM of the game is In this case, H is the dominant strategy for both players, and the SPBM has the unique Nash equilibrium (H, H). The objective payoff allocation given by the solution is not Pareto efficient. By moving from (H, H) to (D, D), each of the players can improve his or her objective payoff.
In the Hawk-Dove game, individualistic and prosocial players achieve Pareto-efficient Nash equilibria. In contrast, competitive players are attracted to a PINE. Similar to the WVG, the Hawk-Dove game is a potential social dilemma.

Battle of the Sexes
The OPBM of the Battle of the Sexes is (F, F) (F, C) (C, F) (C, C) = (2, 1) (0, 0) (0, 0) (1, 2) , where strategies F and C indicate Football and Concert, respectively. If players are individualistic, the SPBM of the game is equal to the OPBM. The SPBM has two Nash equilibria (F, F) and (C, C). The objective payoff allocations given by the solutions are Pareto efficient.
If players are prosocial, the SPBM of the game is which has the same solutions as the game played by individualistic players. If players are competitive, the SPBM of the game is In this case, F is the dominant strategy for player 1, while C is the dominant strategy for player 2. Therefore, (F, C) is the unique Nash equilibrium, which gives a Pareto-inefficient objective payoff allocation.
In the Battle of the Sexes, individualistic and prosocial players achieve Pareto-efficient Nash equilibria, but competitive players are attracted to a PINE. Similar to the WVG, the Battle of the Sexes is a potential social dilemma.

Public Goods game
Assume that each player i ∈ {1, 2} has an initial endowment c i > 0 and provides public goods by paying x i ∈ [0, c i ]. Player i's objective payoff is given by c i − x i + α(x 1 + x 2 ). α is a parameter satisfying 1/2 < α < 1. The OPBM of the Public Goods game is expressed as where strategies C and D indicate Cooperate and Defect, respectively. A player with strategy C contributes to public goods (x i > 0), while a player with strategy D is a free rider (x i = 0). In this game, the strategy profile (C, C) is the solution that maximises the total objective payoff U 1 + U 2 . If players are individualistic, the SPBM of the game is equal to the OPBM. D is the dominant strategy for both players, and the SPBM has the unique Nash equilibrium (D, D).
If players are competitive, the SPBM of the game is where D is the dominant strategy for both players. Similar to individualistic players, (D, D) is the unique Nash equilibrium. If players are prosocial, the SPBM of the game is In this case, C is the dominant strategy for both players, and the SPBM has the unique Nash equilibrium (C, C). In a Public Goods game with a sufficiently high α, the total objective payoff is maximised when both players are prosocial.

Behaviours of altruistic and sadistic players in the WVG
As shown by Figure 1, altruistic and sadistic players are characterised by the weight vectors (α i , β i ) = (0, 1) and (α i , β i ) = (0, −1), respectively. Both of the players have no interest in their own objective payoffs and make decisions based on the other's objective payoff. With an increase in the other's objective payoff, the subjective payoff increases for the altruistic player but decreases for the sadistic player.
If P > 0, the SPBM has three Nash equilibria: (T, F), (F, T), and (F, F). The concentration of tours, denoted by (T, T), does not emerge in the altruistic market. If P = 0, U TT 1 is equal to U TF 1 , and all the strategy profiles are Nash equilibria. The strategy choice of the altruistic player is independent of the game parameters. In other words, the Nash equilibria hold at any point of the parameter region D. From Figure 4, we immediately find that the objective payoff allocations given by the Nash equilibria are not always Pareto efficient.
When players are sadistic, the SPBM of the WVG is If P > 0, the SPBM has two Nash equilibria (T, T) and (F, F). If P = 0, all the strategy profiles are Nash equilibria. Similar to the altruistic player, the strategy choice of the sadistic player is also independent of the game parameters. The objective payoff allocations given by the Nash equilibria are not always Pareto efficient.