Abstract
Game theory studies the choices of two or more agents strategically interacting under various conditions. This paper examines two applications of omniscience in game theory. The first has to do with the paradox of altruism. The paradox of altruism results when players, by seeking to maximize the outcomes of other players, bring about inferior outcomes for all the players. Not surprisingly, an omniscient player could not find herself ensnarled in an altruistic paradox. The second application is what Steven Brams has called the “paradox of omniscience”. The paradox of omniscience is the surprising proposition that omniscience disadvantages a player in certain games if her opponent knows that she is omniscient. Contrary to Brams, I argue that there is no resolution of the paradox of omniscience with regard to normal games. Indeed, I show that omniscience is not the only divine property that is a strategic liability in normal games. Omniscience then has a mixed standing in game theory—in some cases a strategic asset; in others, a strategic liability.
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Notes
The disbursement of benefit has not been entirely one-sided as philosophy of religion has generated areas of discussion that have benefitted other areas of philosophy. Much of the early work in the epistemology of disagreement was in the context of discussions in the philosophy of religion. See, for example, van Inwagen (1996), pp. 137-53. For another example, the issue of what is now called “Pragmatic Encroachment” was presciently discussed by George Maverodes in a 1986 article, years before it became a topic in Epistemology, See Maverodes et al. (1986).
See, for example, Plantinga (1974).
See, for example, Swinburne (1979).
See, for example, Jordan (2008).
Game theory began with the 1944 publication of Theory of Games and Economic Behavior (Princeton University Press) by Neumann and Morgenstern (1944). William Poundstone provides an accessible overview of the development of game theory in Poundstone (1992). As far as I know, Steven Brams was the first to apply game theory to philosophy of religion topics. See, for example, Brams (1982a), (1983), (1980b), (2018). For a critical response to Brams, see McShane (2014), in which she argues that game theory has limited utility in the philosophy of religion. For a more positive assessment of game theory and the philosophy of religion, see Bartholomew (1996), pp. 239–47.
For philosophical uses of game theory in a cooperative setting, see Gauthier (1987).
Brams, Superior Beings, p. 2.
Any strategy associated with an outcome which no player would have an incentive to move away from (unilaterally) as she would do immediately worse, or no better, is a “Nash Equilibrium.”.
McShane, op cit., pp. 9–11.
See J.R. Lucas’s discussion of altruistic games in (1980), pp. 35-71.
While the couple in the O. Henry story end-up with unpreferred material outcomes, those unhappy material conditions manifest their mutual love. So, in that sense, the couple is arguably better-off overall.
I owe this point to Douglas Stalker.
This notion of omniscience may be too wide-ranging, as some have argued that there are truths that even an omniscient being cannot know at certain times. For example, some have argued that there could be truths that cannot be believed, based on the Liar Paradox. See for example, Grim (1991), p. 95; and see Howson (2011), pp. 200–205. Others have argued that there are truths knowable only to some but not all—De Se knowledge for example. Still others have argued that results in Set Theory (Cantor’s Theorem for example) show that there cannot be a set of all truths, so omniscience as knowledge of all and only truths would violate those results—see Patrick Grim, The Incomplete Universe: Totality, Knowledge, and Truth, p. 8. I ignore these complications here.
Consider instead a replacement notion of omniscience: a knower K is omniscient at a time T just in case K knows all the true propositions that it is logically possible for K to know at T, and K believes no false propositions. This definition however may be too constricted, since it would imply that a knower K, who knows only one truth, and is such that K can logically know only that one truth, would be omniscient. Although satisfying the replacement notion of omniscience, it would be odd to consider K omniscient, as too many unknown truths remain on the table.
Some suggest that the two epistemic goals should be calibrated with a third goal of relevance. The appeal of a relevance goal is largely practical as otherwise a knower would waste resources in the pursuit of irrelevant and pointless truths. Inefficiency however is no concern for a limitless knower; and relevancy is superfluous for an omniscient knower.
Brams, Superior Beings, pp. 69–74.
Some notions of omniscience involve a timeless or eternal individual such that that individual, being external or outside of time, does not foreknow anything, even if he knows all truths. An analog of Brams’ paradox of omniscience is constructible even with that complication, since the ordinary player could know that the superior but eternal player timelessly knows that she will choose such and such.
I ignore the problems alleged to plague simple foreknowledge, and the notion of middle knowledge.
If Rock, Paper, Scissors is a normal game in the game-theoretic sense then the issue of pure v. mixed strategy does not arise given Rule R4. It is only regarding multiple plays that the issue arises.
Hasker (1989), p. 188.
Hasker, op cit., p. 192.
An ordinary coin has about 1/6000 chance of landing on its edge. See Murray and Teare (1993).
I am not suggesting that an appropriate value for “significantly high” is as high as in the coin case.
I owe this objection to an anonymous reviewer.
Clearly, acting randomly is not a reliable way to either maximize or even advance one’s preferences, or the preferences of the other player.
Strategies are distinct from preferences in the sense that preferences have to do with outcomes of competing strategies of the players. That distinction is lost if one claims that one’s preference just is to play randomly, as opposed to the relevant outcomes. A relevant outcome would be, for example in the “Vacation Game,” vacationing together at the beach or together in the mountains, or doing so alone.
Many properties will be exploitable if those properties tend to manifest a certain way more often than not, and one’s opponent knows this. The degree to which a property is exploitable varies proportionately to the degree of manifestation. Knowing patterns of behavior displayed by one’s opponent provides a potential strategic advantage.
Brams, (1982c), pp. 17-30.
Are great-making properties logically-preservable, such that any property which follows from a great-making property is itself great-making? I know of no reason to think they are.
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Acknowledgements
I thank Douglas Stalker and Noel Swanson for their helpful suggestions. An anonymous reviewer provided valuable questions and informative comments.
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Jordan, J. Game theory and omniscience. Int J Philos Relig 94, 91–106 (2023). https://doi.org/10.1007/s11153-023-09866-1
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DOI: https://doi.org/10.1007/s11153-023-09866-1