Kennedy Goodkey

December 24, 2002

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Does the key to winning Survivor lie in an understanding of non-zero sum games? Jeff Probst seems to think so, as he mentioned at the end of Survivor: Thailand. But what was he talking about when he referenced John Nash's "Non-Cooperative Games"? What does it mean and how does it work?

I've been casually interested in game theory for a few years now - not quite as long as I've been a *Survivor* addict. I suspect that there is a correlation between the two. I've long since held that the key to winning *Survivor* lay in a sharp critical understanding of non-zero sum games without perfect information for more than two players.

Huh?

Well Jeff Probst let the cat out of the bag at the end of the reunion special on Thursday night. Too bad. I've been quietly keeping that to myself for the day that eligibility rules change and Canadians are able to participate in *Survivor*. So before anyone gets to demonstrate it on the show, I thought I better take my opportunity to illuminate the details in my best layman's terms.

First a disclaimer: I am not a mathematician - and anything I've learned on the subject after high school is entirely self-taught. So anything I say could well be subject to correction by someone with a real education.

Jeff recommended John Nash's "Non-Cooperative Games." If the name is familiar, it's because John Nash is the real mathematician who Russell Crowe played in last year's Best Picture Oscar winner, *A Beautiful Mind* (you can read the NonfictionReviews.com review of the book on which the movie was based by clicking here). The paper (which can be found, along with others, in the book, *The Essential John Nash*) is the basis for which he was awarded the Nobel Prize. Mathematically speaking it is a revolutionary work of science. It would not be unfair to say that its impact is akin to Darwin's "Origin of Species" or Einstein's "Theory of Relativity."
But in fact, there is a LOT more to Game Theory that could be of use in *Survivor*.

For example: In episode six of Thailand, the immunity challenge was a game called Thai 21. This is one of the most basic of mathematical games, one that would come up very early in any study of game theory. If each team had a player with the right mathematical skill, the first team to move would have won the game - every time. This did not happen. You may recall, there were 21 flags and each team in order had the opportunity to remove 1,2, or 3 flags. The team to remove the last flag won immunity.

Sook Jai relinquished control by leaving Chuay Ghan with ten flags when they could have left them with twelve. I've long since erased the tape I had it on, but I recall thinking, "Oh, there - they just blew it. And sure enough Chuay Ghan and Sook Jai both soon noticed that the game was inevitably going to Chuay Ghan. How so?

A team with an unbeatable strategy will always leave the opposing team with a multiple of the maximum number of flags that a team can take plus one. In this way the opposing team cannot avoid leaving a savvy controlling team in control. In the case of Thai 21, the team with control would leave the opponent with 20 flags the first turn. No matter how many flags were then taken by the opponent, the team in control could then leave the opponent with 16 flags and then 12 and then 8 and lastly 4. Left with only four flags and the requirement to take at least one, but not more than three, means that you MUST put the opponent within range of winning, but cannot win yourself. It is a 'master-move.'

But this example game is child's play compared to the intricacies of *Survivor*'s uber-game... Tribal Politics!

Nash's greatest feat of intellectual achievement is the formulation of the Nash Equilibrium. The Nash Equilibrium is a wonderfully simple idea, but very difficult to master in practice. In fact, the basic idea is SO simple that you'll think, "Well, THAT is obvious." But keep in mind, it's also easy for us to understand that the Earth goes around the Sun, not the other way around. Before Copernicus it seemed illogical to accept anything other than the Ptolemaic view that the Earth was the center of the Universe and that everything else surrounded it.

Let's take this one step at a time.

In any game of strategy, the outcome hinges not only on what YOU as a player do, but what every other player does. Furthermore, this applies to each player individually.

In a two-player game where both players' moves and options for moves are available to each and each player moves in sequence, there is a fixed number of possible options. In a game like Tic-Tac-Toe, or even Thai 21, the various options are relatively easy to calculate and react to. In Chess, things get more complicated, but essentially the principle of looking ahead in the game to determine the best possible strategy based on a limited number of options and no hidden information remains the same. The other player's specific strategy may be unknown, but you always know what the situation on the board is - what pieces are in play, what options are available.

But *Survivor* and Chess have very little in common. Unless you're playing some new version of Chess where there are sixteen players, none of them actually have their pieces on the table, all players move whenever they feel like it, at regular intervals players are forced to checkmate someone but at any given time there's at least one player you aren't allowed to checkmate, and players are allowed to make agreements behind each other's backs. Okay, no matter how you cut it, Chess is nothing like *Survivor* - people should quit making naïve comparisons to it.

The politics of *Survivor* is in an entirely different order of game, involving the bluffing of Poker, the cooperative bargaining of Monopoly, the tactical scheming of Risk, and cutthroat opportunism of Sorry - amongst many other elements, physical, intellectual, and emotional. In short, it's very complex. But above all else it still maintains the primary element of interdependency. Every player's best tactics are affected by every other player. Furthermore, knowing that each player is interdependent changes your tactics AND changes their tactics ad infinitum. (E.g. "I know that she knows that I know that she knows that I know...") You could lose your mind just trying to get to the base of it. It is circular as reasoning can get.

So, how does one simplify this? Well, I think it is safe to say that there is no 'master-move' in *Survivor*. Like with Rock/Paper/Scissors, there is always a way to beat any one strategy. So the best you can ever hope for is to maximize your chances by picking your best move in the situation - a situation where you can't possibly know, let alone calculate all the possible variables.

This is where the Nash Equilibrium comes into play. It's certainly fallible, but the player who best puts it to work will, all else being equal (yeah, right!) win *Survivor*.

To put the strategy to work you must understand that there is a point (an equilibrium) where every player's choices maximize their individual outcome in the game - or rather, at the next stage. For ease you can divide the game into fifteen stages - the fifteen Tribal Councils. But in reality there are more subdivisions within this which also need to be considered along the way, ("Do I help go for water, or do I conserve energy?" "Do I throw this immunity?" Etc.)

Simply because every player's outcome is maximized does NOT mean that every player gets the best conceivable result. Otherwise there would now be 80 different million-dollar winning 'Sole Survivors' rather than five. What it does mean is that in every circumstance there is a 'best-solution' for each player that is totally dependent upon the other players, but entirely mutable by a canny strategist.

How does any one player make themself that 'canny strategist' and maximize the result by picking their 'best-solution?' First they must start by eliminating the variables that do not change in any given situation. In any circumstance this is always a best-guess type scenario. The players can only ever depend upon what they know or at least what they suspect strongly enough to take a chance upon.

In a given situation it is possible that there are players whose choices are going to be the same regardless of the other players' choices (called a dominant strategy). There are also players whose situations cannot be improved no matter what choice they make (a dominated strategy). Both of these types of players can effectively be eliminated from all other players' reasoning - as their results are fixed. They have no immediate impact upon the potential fluctuations of the other players, so once you've identified a player in that position, you don't need to confuse yourself further with how they will act and by extension how minute changes in the overall group affect them or vice versa.

From this point all you need to determine is the equilibrium point. Where will the cards lay when each player has determined to the best of their knowledge what they must do to put themselves in a position where they could not improve their situation by choosing any other available choice. Armed with this knowledge, you can exploit it. In a fashion not unlike Heisenberg's Uncertainty Principle - knowing the state of the situation changes it. Luckily, in the game of *Survivor*, knowledge is power.

This is but one way in which game theory can help a *Survivor* player. There are many more ways which can be explored later - such as another work of John Nash; his solution to "The Bargaining Problem" (also included in *The Essential John Nash*), but that's another story...

Game Theory *.net*

© Mike Shor 2001-2006

© Mike Shor 2001-2006