While thinking some things over in my head, I came across the idea of trying to generalize the Prisoner's Dilemma to multiple participants, and trying to do it in the simplest way possible.

In essence, what I came up with is that between each ordered pair of players, there is the action of "take 1". Each player can either do it or not do it to another player. If the player does it, he gains one point, while the person whom he "took 1" from loses two points.

In the case of two players, each with the action "take 1" or "don't take 1", the payoff matrix looks like the traditional Prisoner's Dilemma [transpose for player 2]:

$\begin{matrix} \text{Player 1} & \text{Take} & \text{Don't} \\ \text{Take} & -1 & +1 \\ \text{Don't} & -2 & 0 \\ \end{matrix}$

(It's a nonpositive-sum game because people are usually more loss-averse than they are gain-prefering, so removing global gain as an option generally forces more psychological strain [and often more irrational, destructive action] on the players, which is part of the point of the Prisoner's Dilemma.)

It generalizes in the global behaviour of the Nash equilibrium - "take 1" is still the more locally beneficial ("personally rational") option, but leads to the worst global result.

My question is, does such a generalization (where the dilemma is split into multiple independent actions) already exist, and whether studies have been done on it. If so, where can I find it (and its analysis)? If not, how could I go analyzing such a multiplayer situation myself?

  • $\begingroup$ Considering that you know what you want, do a search for N-player Prisoner's Dilemmma. It does exist (not new), and there has been a bunch of research on it. $\endgroup$ – Calvin Lin Dec 30 '12 at 0:45
  • $\begingroup$ Oh, I know there are lots of multiplayer generalizations. What I meant was, does my specific sort of generalization (where the dilemma is split into independent actions) exist. $\endgroup$ – Joe Z. Dec 30 '12 at 1:33

I am not sure that I fully got your idea (I don't quite get why the game is played between each ordered pair of players), but at the first sign it looks like that you came up with a rather natural (and brilliant) idea of replacing playing mixed strategies by two players with the idea to consider a symmetric game where each player of population plays a pure strategy, and the structure of population changes according to the individual outcomes of the game. This theory is called evolutionary game theory and has tons of research and applications (in biology, but also as far as I know in economics). In particular, if your payoff matrix is $A$ and it is assumed that the payoff of the game is identified with the biological notion of fitness (the reproductive success), then the evolution of the subpopulations playing various pure strategies can be described with the help of the so-called replicator equation: $$ \dot p_i=p_i(\sum_j a_{ij}p_j-p\cdot Ap),\quad i=1,\ldots,n, $$ where $p\in S_n=\{p\in\mathbf R^n\mid p\geq 0\,, \sum p_i=1\}$ and $p\cdot Ap$ denotes the usual inner product (you need this to keep the simplex $S_n$ invariant).

  • $\begingroup$ "(I don't quite get why the game is played between each ordered pair of players)" $\leftarrow$ No, between each ordered pair of players there is either an action or no action. So each turn can be represented as a simple directed graph where each player is a vertex. Basically, each player can either take 1 or not take 1 from each other player, and this represents itself as "each ordered pair of players can be either 'take 1' or 'not take 1'". $\endgroup$ – Joe Z. Dec 30 '12 at 3:01
  • $\begingroup$ But thanks for the explanation. The symmetrization of the game is exactly what I had in mind. $\endgroup$ – Joe Z. Dec 30 '12 at 3:02
  • $\begingroup$ Ok, I see now. What I described above is for a complete graph (each player has a chance to play with any one). But of course generalizations are possible. If you are still interested in this sort of things, look through this book. It has a lot of pertinent references. $\endgroup$ – Artem Dec 30 '12 at 3:04

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