Survivor, Game Theory, and John Nash

At the very end of the Survivor 5 reunion show, Jeff Probst made a quick, cryptic remark about a theory to explain how to optimally play Survivor. As an academician who actually has published in this area (bet nobody can find the name of that piece), I feel it's my duty to shed a bit more light on this remark, especially since most of my columns deal with Survivor from an international relations/game theory perspective.

He referred to the now-famous Nobel Prize winner John Nash, popularized in the movie A Beautiful Mind (you can read the NonfictionReviews.com review of the book on which the movie was based by clicking here), who some years ago postulated a theory to better understand outcomes in uncooperative (that is, competitive) games that are nonzerosum as well as zerosum (depending upon your point of view, Survivor can be either, but I'm not going to get into that esoteric argument unless a deluge of mail requests me to). To relate it to Survivor, it argues that there is at least one equilibrium solution (all players able to pursue a fixed strategy) where payoffs will differ depending upon the ability of each player to employ rationality at each step of the way.

So even if competition exists, this is why alliances get made to further individual interests. To rephrase this in terms of how the game is supposed to be played, first, each player chooses a threat, or what he will be forced to do if they can't make a deal, that is, if their demands are incompatible. Next, the players tell each other the threats. Then, each player chooses a demand that is an outcome worth a certain amount to him. If the bargain does not guarantee him that amount, he will not agree to a deal. Finally, if they can settle on a deal that satisfies both players' demands, the players will get what they are asking for. If not, they will have to carry out their threats. Nash showed that a unique stable equilibrium exists that coincides with the bargaining solution in which each player has an "optimal" threat that ensures that a deal will be struck, no matter what strategy the other player chooses, and this is how the negotiation process has been structured (this phrasing courtesy of Nasar's biography of Nash).

Thus, each player, given his strengths and those of others, can play any number of ways a set strategy. Presumably, for each player there exists at least one optimal strategy, or one that promises to bring the player the farthest in the game. That also implies that for one player, if he discovers and employs this strategy without error, will win the game regardless of what others do (a "fixed" strategy). If he does not, that opens the door for another player to win. This means that all players can win, but some are more likely to do so than others. In other words, there is established a point where no party can be better off when they change their strategy, but they could be better off if all of them would change their strategy.

In essence, this means that alliances get made, or not made, because each participant believes it will get him as far as he can would be satisfied in going. They might not mind going further (finishing closer to or at the top) but the strategy they choose at least assures them of this minimally satisfactory outcome. This implies that Nash's equilibrium does produce a predictable outcome, given the various desires for each player, but assuming that rationality always is employed.

But Probst alluded that a viewer could look at all 16 players, or a player at himself and his competitors, and could figure out who would win. Naturally, nobody has such perfect knowledge of all players' minimally-acceptable end-states (and, further, luck may interfere - see Mike Skupin in S2), so from the individual player's perspective, he must determine his optimal strategy and not worry about what others do. In that sense, Probst acted disingenuously when suggesting one could figure out the whole game, implying a player could win knowing this, by knowledge of Nash's theory of equilibrium. All one can do is employ rationality (whether perfect or "bounded" - reasonably close to perfect - is yet another request to explain only if prompted through a deluge of mail) maximally and hope the fixed strategy that results is good enough to make you a millionaire.

To use S5 as an example, one could assert that the players who went the farthest did adopt fixed strategies (again, they consistently follow a certain set of decision rules) making alliance deals to achieve their minimums. For Brian, he decided to tell people what they wanted to hear, hoping to be convincing enough that they did not question him or themselves to discover that he had set up a set of mutually incompatible interlocking alliances, execution of which would win jury admiration. For Clay, he chose to be himself, hoping that bluntness could create a perception of irascibility that would make others think him to be jury fodder but which in reality would be seen as refreshing compared to other strategies and could win him the jury vote with the right opponent. For Jan, given her strengths and weaknesses, she decided to tag along with what she correctly perceived as the dominant alliance and then hope for luck to get her the win. One could argue that the top three finished where they did because that was what they wanted at a minimum (although it could not possibly have been lower except in Brian's case; since we assume players combine to finish at the minimum acceptable level when behaving rationally, there is nothing else acceptable to the player whose minimal acceptable outcome is to win).

Even Helen's fourth place can be interpreted consistently with the theory. While surely she hoped for more, given her capabilities (including an inability to see past Brian's machinations), in the back of her mind when allying with him she must have known the risk and have been satisfied with a non-winning hand should her trust get betrayed. Her case illustrates that Nash's theory can be used in a post-hoc sense to understand game outcomes, but in and of itself does not suggest a particular strategy a player should pursue. As I have argued elsewhere (for example, here and here and here), there were strategies I believe could have brought her a lot closer to victory, but, as the theory suggests, perhaps she did not pursue these strategies in part because her minimum satisfactory outcome was at least fourth place and feared taking greater risks to lose that minimum.

It is unrealistic to believe that understanding game theory generally and Nash's theory in particular will make you a winner of or able to predict the winner of Survivor, but it does help one understand the play of the game and, as a contestant, knowledge of it in pursuing a strategy gives you an edge (sounds like skills needed of a successful used-car salesman, doesn't it?). This theory comes from the mathematics discipline but Survivor is more a game of politics, and, as I have argued, specifically international relations. So, I close here with a short bibliography of works in political science that one may pursue to get a better handle on the dynamics of behavior behind Survivor:

• Conflict of Interest; a Theory of Divergent Goals with Applications to Politics, by Robert M. Axelrod. Provides a general theoretical overview specifically related to politics, more at the individual than systemic level, the latter level where some sort of governing authority is assumed to be present or desired to be implemented which is not the case in Survivor except for the jury concept.
• Game Theory and Political Theory : An Introduction, by Peter C. Ordeshook. This is a more specific theoretical treatise connecting political theory (in an oversimplified sense, how to manage social conflict) to game theory at the individual level.
• Theory of International Politics, by Kenneth N. Waltz. This work specifically looks at, in part, game theory applications in an anarchic yet subconsciously-organized environment most closely resembling the environment provided in Survivor. The drawback here is that Waltz concentrates more on the highest (system) level of analysis and the jury system imposes an element of structure basically absent in the real world of international relations.

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Jeffrey D. Sadow is an associate professor of political science at Louisiana State University in Shreveport where he teaches, among other things, classes in international politics, international organizations, and diplomatic history. He has published in the area of gaming simulations in international politics.

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