Abstract
Democracy resolves conflicts in difficult games like prisoners’ dilemma and chicken by stabilizing their cooperative outcomes. It does so by transforming these games into games in which voters are presented with a choice between a cooperative outcome and a Pareto-inferior noncooperative outcome. In the transformed game, it is always rational for voters to vote for the cooperative outcome, because cooperation is a weakly dominant strategy independent of the decision rule and the number of voters who choose it. Such games are illustrated by 2-person and n-person public-goods games, in which it is optimal to be a free rider, and a biblical story from the book of Exodus.
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- 1.
The Nash equilibrium is actually the pair of pure strategies of the players associated with (2,2), not the outcome itself, but for convenience we identify Nash equilibria by the outcomes they produce. We do not consider mixed strategies in this or other 2 ×2 games that we discuss later, because preference information is in ordinal rankings, rather than cardinal utilities, precluding expected-utility calculations that underlie the determination of mixed strategies.
- 2.
Farsightedness offers a very different resolution of PD than tournament play or evolution. Pinker (2007, p. 71) distinguishes the former from the latter by arguing that “natural selection [in evolution] is like a design engineer in the sense that parts of animals become engineered to accomplish certain things, but it is not like a design engineer in that it doesn’t have long-term foresight.” Presumably, only humans possess this foresight and can anticipate that if they move from (3,3), it will not necessarily induce their best outcome of (4,1) or (1,4) but, instead, may trigger a countermove by the player receiving 1 to (2,2). Because this outcome is worse for both players than (3,3), (3,3) is a “nonmyopic equilibrium” in PD if the players start at this outcome and think ahead (Brams and Wittman 1981; Kilgour 1984; Brams 1994).
- 3.
As in PD, the cooperative outcome in chicken is a nonmyopic, but not a Nash, equilibrium. In fact, the game that best models this crisis, and its resolution, is probably neither PD nor Chicken but a different game (Brams 1994, pp. 130–138).
- 4.
This equilibrium is Pareto-optimal, or efficient, because no other outcome is better for both players than (3,3). Although the lower-right (2,2) outcome is also a Nash equilibrium, and therefore stable, the players would have no reason to choose it over the (3,3) outcome, to which it is Pareto-inferior.
- 5.
Why “weakly”? Unlike PD, each player’s cooperative strategy associated with (3,3) is not strictly better, whichever strategy the other player chooses: If the other player votes not to finance, either voting to finance or voting not to finance leads to the same outcome of (2,2). Because of this “tie,” voting to finance is not always better than voting not to finance.
- 6.
Because c(k) and d(k) are indexed differently, we can compare c(k) and d(k − 1) over all k, as we do in property (2) below.
- 7.
In the preceding example, we treated the “rest of the public” as a single player, but if the game is among many similar players, then it is properly modeled as an n-person PD. To ameliorate the problem of defections in such a game, wealthy individuals often commit to match the donations of small contributors, thereby enhancing the incentive of these individuals to contribute by guaranteeing that their donations will be increased by some factor.
- 8.
Hardin (1971) shows that all-C is a Condorcet choice when pitted against any other strategy combination – that is, a majority of voters would prefer it, except in the case of a tie – but he does not provide a procedure that would implement all-C.
- 9.
This story is adapted from Brams (1980, 2003, pp 94–98), but the interpretation of Moses’s resolution of an n-person PD via a kind of referendum is new. Passages from the Bible are drawn from The Torah: The five books of Moses (1962). Schelling (1978, ch. 7) gives several contemporary examples of n-person PDs, such as whether a hockey player should wear a helmet, which was not mandated by the National Hockey League (NHL) until the 1990s. Prior to 1990, most players refused to wear helmets because it put them at a strategic disadvantage, limiting their peripheral vision, though they were at a substantially greater risk of serious head injury. The dilemma was resolved not by a secret vote of the players, which arguably would have led to the requirement of helmets in the 1970s, but by a public outcry caused by head injuries, which put pressure on the NHL. Even so, players who entered the league before the helmet requirement were exempted; the last player to refuse to wear a helmet retired in 1997.
- 10.
The privacy of a voting booth is important if voters might be under social pressure to vote differently if their votes were known. To be sure, this social pressure might be critical to the passage of certain kinds of legislation, such as that backed by a political party that can punish defectors when there is a roll call vote. Perhaps the support that Moses, who was a Levite, received from his fellow Levites was reinforced by the public nature of those rallying to his side gave him. By contrast, the ringleaders on a ship who pledge in writing to participate in a mutiny are immediately identifiable, and subject to severe punishment, if they are discovered before the mutiny and were the first to sign the pledge. The institutional solution that mutineers devised to prevent the discovery of the ringleaders was to write their names in a circle (“round robin”) (Leeson, 2007).
- 11.
If we include games with (4,4) outcomes, there would be 21 additional games, making for a total of 78 distinct 2 ×2 strict ordinal games (Rapoport and Guyer 1966,1976); see Robinson and Goforth (2005) for a further elaboration of these games and their properties. We exclude the games with (4,4) outcomes, because these outcomes are the unique Pareto-optimal Nash equilibria in them, rendering (4,4) the likely outcome that players would choose without the need for voting. The one exception is a game variously referred to as stag hunt, assurance, or coordination (Skyrms 2004):
Table 4 If either player in this game chooses its second strategy, it assures itself of a minimum of 2, whereas choosing its first strategy may lead to 1. Thus, a player’s second strategy is, in a sense, less risky; its choice by both players yields (2,2), which is a Nash equilibrium, albeit Pareto-inferior to (4,4).
- 12.
Schelling (1978, ch. 7) offers a different classification of PD and non-PD games, using lines and curves on a graph. Still other classifications of the 78 2×2 strict ordinal games, which include the 57 games of conflict and 21 games with a mutually best (4,4) outcome, are given in Rapoport and Guyer (1966,1976) and Brams (1977); a topology of such games, and even a new classification in a “periodic table,” are developed in Robinson and Goforth (2005).
- 13.
Beginning in the 1960s, the United States and the Soviet Union were able to reach limited arms-control agreements, because both asides could detect violations of these agreements with a sufficiently high probability (e.g., via satellite reconnaissance) and take appropriate countermeasures if a violation were detected. By and large, this deterred both superpowers from violating these agreements.
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Acknowledgment
We thank Todd R. Kaplan, Christian Klamler, Maria Montero, Brian Skyrms, and Donald Wittman for valuable comments on an earlier version of this paper.
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Brams, S.J., Kilgour, D.M. (2009). How Democracy Resolves Conflict in Difficult Games. In: Levin, S. (eds) Games, Groups, and the Global Good. Springer Series in Game Theory. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85436-4_14
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