Game theory is a framework for analyzing the outcome of the strategic interaction between decision makers1. The fundamental concept is that of a Nash equilibrium where no player can improve his payoff by a unilateral strategy change. Typically, the Nash equilibrium is considered to be the optimal outcome of a game, however in social dilemmas the individual optimal outcome is at odds with the collective optimal outcome2. This means that one player can improve his payoff at the expense of the other by unilaterally deviating, but if both deviate, they end up with lower payoffs. In this type of games, the mutually beneficial, but non-Nash equilibrium strategy is called cooperation. However, in this context cooperation should not be interpreted as an interest in the welfare of others, as players only aim to secure a high payoff for themselves.

In this framework, payoff maximization is considered to be rational, but when such rational players then seize every opportunity to gain at the opponent’s expense, they may counterintuitively both end up with low payoffs. A game that clearly exhibits this contradiction is the Traveler’s Dilemma. Since its formulation in 1994 by the economist Kaushik Basu3, the game has become one of the most studied in the economics literature. Additionally, it has been discussed in theoretical biology in the context of evolutionary game theory.

In general, the dilemma relies on the individuals’ incentive to undercut the opponent. To be more specific, players are motivated to claim a lower value than their opponent to reach a higher payoff at the opponent’s expense. Such incentive leads players to a systematic mutual undercutting until the lowest possible payoff is reached, which is the unique Nash equilibrium. It seems paradoxical that players defined as rational in a game theoretical sense end up with such a poor outcome. Therefore, the question that naturally arises is how this poor outcome can be prevented and how cooperation can be achieved.

To address these questions, it can be helpful to better understand price wars, which consist in the mutual undercutting of prices to gain market share. In addition, it can provide information about human behavior, because economic experiments have shown that individuals prefer to choose the cooperative high payoff action, instead of the Nash equilibrium4.

Our analysis focuses on showing that the Traveler’s Dilemma can be decomposed into a local and a global game. If the payoff optimization is constrained to the local game, then players will inevitably end up in the Nash equilibrium. However, if players escape the local maximization and optimize their payoff for the global game, they can reach the cooperative high payoff equilibrium.

Here, we show that the cooperative equilibrium can be reached in a game like the Traveler’s Dilemma due to diversity, which we define as the presence of suboptimal strategies. The appearance of strategies far from those of the residents allows for the local maximization process to be escaped, such that an optimization at a global level takes place. Overall this can lead to cooperation because by considering “suboptimal strategies” that play against each other it is possible to reach higher payoffs, both collectively and individually.

### Game

The Traveler’s Dilemma is a two-player game. Player i has to choose a claim, (n_i)from the action space, consisting of all integers on the interval [LU]where (0 le L < U). The payoffs are determined as follows:

• If both players, i and jchoose the same value ((n_i = n_j)), both get paid that value.

• There is a reward parameter (R>1)such that if (n_i < n_j)then i receives (n_i + R) and j gets (n_i- R)

Thus, the payoff of the player i playing against players j is

begin{aligned} pi _{ij} = {left{ begin{array}{ll} n_i& text { if } n_i = n_j\ n_i + R& text { if } n_i < n_j n_j - R& text { if } n_i > n_j end{array}right. } end{aligned}

(1)

Thus, a player is better off by choosing a slightly lower value than the opponent: when j plays (n_j)then it is best for i to play (n_j-1). The iteration of this reasoning, which we will call the stairway to hellleads to the only Nash equilibrium of the game, ({L,L}), where both players choose the lowest possible claim. The classical game theory method to reach this equilibrium is called iterative elimination of dominated strategies5.

The game can be visualized through its payoff matrix (Fig. 1). For simplicity, we use the values ​​from the original formulation: (L=2), (U=100) and (R=2). The payoff matrix shows that the Traveler’s Dilemma can be decomposed into a local and a global game. Let us begin with the local game. When the action space of the game is reduced to two adjacent actions n and (n+1) (black boxes in Fig. 1), the Traveler’s Dilemma with (R=2) is equivalent to the Prisoner’s Dilemma6. In general, for any value of Rthe Traveler’s Dilemma becomes a Prisoner’s Dilemma for any pair of actions n and (n+s)where ( 1 le s le R-1 ). For example, for (R=4) the pair of actions n and (n+1), n and (n+2), n and (n+3) follow the same game structure as the Prisoner’s Dilemma. Therefore, the Traveler’s Dilemma consists of many embedded Prisoners’ Dilemmas. This means that at a local level the game is a Prisoner’s Dilemma.

If we now consider actions that are distant from each other in the action space, eg 2 and 100, we can observe a coordination game structure (gray boxes in Fig. 1), where ({100,100}) is payoff and risk dominant7,8. In general, any pair of actions n and (n+s)where ( R le s le Un), construct a coordination game. As a result, the Traveler’s Dilemma becomes a coordination game at a global level, which has different equilibria than the local game.