You might be familiar with game theory and Nash equilibria. Even if you are not, you probably know of the eponymous and troubled John Nash (the subject of the movie A Beautiful Mind).
Game theory is not about games, it’s about modeling decisions. Nash equilibria are useful for finding stable solutions to “games” have very specific criteria. You have participants employing strategies that they cannot change, they know each others choices (and outcomes given a combination of strategies), and the players are rational.
A simple but relevant example is perhaps a pair of candidates are competing in a primary. The winner will face the other party’s candidate in a general election. Both candidates in the primary can opt to run positive or negative campaigns and each choice results in different payoffs for each candidate relating to the odds in the general/down ballot success (the specific payoffs of the outcomes isn’t relevant, only that that they are correct relative to each other).
For simplicity’s sake, let’s say they are symmetric. If both candidates run a positive campaign, the odds are higher that the winner will take the general and that the candidates party will succeed down ballot (stronger voting block). So, on the whole, it is a 10 for both not knowing the outcome. If one candidate is negative and the other is positive, the negative candidate will win, but at the cost of a worse general/down ballot result. If both go negative, the result is worst overall.
Below is the payoff matrix.
|
POSITIVE |
NEGATIVE |
POSITIVE |
10,10 |
3,6 |
NEGATIVE |
6,3 |
0,0 |
The way you find the Nash equilibrium is pretty simple, maximize your payoff given the opponents choices.
In this case, two positive campaigns is the equilibrium, because neither player can improve their outcome by switching strategies.
Contrast this with the prisoner’s dilemma (two prisoners are accused of a crime. If both remain silent (cooperate) both get a short prison sentence, if one rats out the other the other while the other remains silent the betrayer goes free and the other serves a long prison sentence, and if they both betray each other they get a medium prison sentence. The no/short/medium/long sentences have payoffs of 0,3,6,10 respectively.
|
cooperate |
betray |
Cooperate |
6,6 |
0,10 |
betray |
10,0 |
3,3 |
This has an equilibrium of both betraying each other, since cooperate/cooperate is unstable (opposing player can improve their outcome by betraying, 10 > 6) given the assumptions of the game. What causes some people trouble is that it’s mutual cooperation is the best overall strategy (Pareto optimal).
Anyway, what is more interesting about this is either when “player” don’t follow the expected strategy. Either they aren’t rational or have different utilities of payoffs than you expected (utility is a measure of how much you “like” something … how much would you pay for a 1/100 chance to win $100? The answer divided by one is your utility of risk, in a basic sense).
Back to the case of the primary, you can note that the negative/positive strategies have asymmetrical outcomes — at the cost of reduced general success, one has tilted the odds in their own favor. Even moreso than the even odds of positive/positive strategies. If someone values winning more than mutual success, the negative/positive has a higher payoff.
When that happens it has the same Nash equilibrium as Prisoner’s Dilemma, and you get the same non-Pareto optimal result.