In this post, I present a generalization of Nash equilibria to nonCDT agents. I will use this formulation to model mutual cooperation in a twin prisoners’ dilemma, caused by the belief that the other player is similar to you, and not by mutual prediction. (This post came mostly out of a conversation with Sam Eisenstat, as well as contributions from Tsvi BensonTilsen and Jessica Taylor)
This post is sketchy. If someone would like to go through the work of making it more formal/correct, let me know. Also, let me know if this concept already exists.
Let \(A_1,\ldots,A_n\) be a finite collection of players. Let \(M_i\) be a finite collection of moves available to \(A_i\). Let \(U_i:\prod_i M_i\rightarrow\mathbb{R}\) be a utility function for player \(A_i\).
Let \(\Delta_i\) be the simplex of all probability distributions over \(M_i\). Let \(V_i\) denote the vector space of functions \(v:M_i\rightarrow\mathbb{R}\), with \(\sum_{m\in M_i}v(m)=0\).
A vector given a distribution in \(p\in\Delta_i\) and a vector \(v\in V_i\) we say that \(v\) is unblocked from \(p\) if there exists an \(\varepsilon>0\), such that \(p+\varepsilon v\in\Delta_i\). i.e., it is possible to move in the direction of \(v\) from \(p\), and stay in \(\Delta_i\).
Given a strategy profile \(P=(p_1,\ldots,p_n)\) in \(\prod_{j\leq n} \Delta_j\), and a vector \(V=(v_1,\ldots,v_n)\) in \(\prod_{j\leq n} V_j\), we say that \(V\) improves \(P\) for \(A_i\) if
\[\lim_{\varepsilon\rightarrow 0}\frac{\sum_{m_1\in M_1}\dots\sum_{m_n\in M_n}U_i(m_1,\dots,m_n)\left(\prod_{j\leq n}p_j(m_j)+\varepsilon v_j(m_j)\prod_{j\leq n}p_j(m_j)\right)}{\varepsilon}>0.\]
We call this limit the utility differential for \(A_i\) at \(P\) in the direction of \(V\).
i.e., \(V\) improves \(P\) for \(A_i\) if \(U_i\) is increased when \(P\) is moved an infinitesimal amount in the direction of \(V\). (not that this is defined even when the vectors are not unblocked from the distributions)
A “counterfactual gradient” for player \(A_i\) is a linear function from \(V_i\) to \(\prod_{j\leq n} V_j\), such that the function into the \(V_i\) component is the identity. This represents how much player \(A_i\) expects the probabilities of the other players to move when she moves her probabilities.
A “counterfactual system” for \(A_i\) is a continuous function which takes is a strategy profile \(P\) in \(\prod_{j\leq n} \Delta_j\), and outputs a counterfactual gradient. This represents the fact that a player’s counterfactual gradient could be different depending on the strategy profile the game ends up in. We fix a counterfactual system \(C_i\) for each player.
Claim: There exists a strategy profile \(P=(p_1,\ldots,p_n)\) in \(\prod_{j\leq n} \Delta_j\) such that for all players \(A_i\), if \(v_i\in V_i\) is unblocked from \(p_i\), then \(C_i(P)(v_i)\) does not improve \(P\).
I will call such a point an entangled equilibrium.
Proof Sketch: (Very sketchy, and I have not verified this. There is probably a better way. It might not be true.)
We will construct a continuous function from \(\prod_{j\leq n} \Delta_j\) to itself. We do this by moving \(p_i\) by adding the gradient of function from \(p_i\) to to \(U_i\), assuming all other players probabilities change according to the linear function \(C_i(P)\). If adding this gradient would take the point out of the simplex \(\Delta_i\), you hit a boundary after moving some proportion \(\alpha\) of the gradient. Then, now that you are on the boundary, you use move by \(1\alpha\) times the gradient of the same function restricted to the boundary, and repeat.
This function has a fixed point, by Brouwer. For each player, this fixed point must have 0 gradient, or be on the boundary, pointing outward. If it is on a boundary, it has the same property when restricted to that boundary.
If there was an unblocked direction a player could move that would improve its utility, then the gradient would be nonzero. If the player is on a boundary, the gradient is pointing outward, and there is an unblocked direction that would be an improvement, there will be such a direction that stays on that boundary, and the gradient restricted to that boundary would be nonzero. \(\square\)
Example 1
Consider a twin prisoners’ dilemma game. Two players can either cooperate or defect. They get 0 utility for being exploited, 3 for exploiting, 2 for mutual cooperation, and 1 for mutual defection.
Both players believe that the other is using a decision procedure that is entangled with their own, but they are not completely sure.
Formally, both players think that when they increase their probability by \(\varepsilon\), the other player increases by \(2\varepsilon/3\), regardless of where the probabilities start.
Thus \(C_1(p,q)(v_1)=(v_1,2v_1)/3\), where \(v_1\) is the vector (\(A_1\) cooperates)(\(A_1\) defects) in \(V_1\), and \(v_2\) is the analogous vector for player 2. Similarly, \(C_2(p,q)(v_2)=(2v_2/3, v_2)\).
Since both players always think that increasing their probability of cooperation increases utility, the only equilibrium is when both players cooperate with probability 1.
Example 2
Now let look at a case where the gradients are a function of the distributions. Again, both players believe that the other is using a decision procedure that is entangled with their own, but they are not completely sure.
Formally, player 1 thinks that when they increase (or decrease) their probably of cooperation by \(\varepsilon\), the other player will increase (or decrease) their probability by \(\delta\varepsilon\), where \(\delta\) is one minus the square of the difference between their two probabilities.
Player 2 on the other hand, thinks the other player will increase (or decrease) their probability by \(\delta\varepsilon\), where \(\delta\) is one minus the absolute value of the difference between their two probabilities.
Thus \(C_1(p,q)(v_1)=(v_1,(1(pq)^2)\cdot v_2)\), where \(v_1\) is the vector (\(A_1\) cooperates)(\(A_1\) defects) in \(V_1\), and \(v_2\) is the analogous vector for player 2. Similarly, \(C_2(p,q)(v_2)=((1pq)\cdot v_1, v_2)\).
Thus, if player 1 cooperates with probability \(p\), and player 2 cooperates with probability \(q\), then player 1 expects to lose gain \((2(1(pq)^2)1)\varepsilon\) utility by increasing his probability by \(\varepsilon\). This function is only 0 if \(pq=1/\sqrt{2}\). Thus the only entangled equilibria with mixed strategies for player 1 have \((pq)=1/\sqrt{2}\). Similarly, the only entangled equilibria with mixed strategies for player 2 have \(pq=1/2\). Thus, there are no mixed strategies for both players in equilibria.
The only pure equilibrium is \(p=q=1\)
If \(p=0\), then \((2(1(pq)^2)1)\) must be nonpositive, so \(q\) must be at least \(1/\sqrt{2}\). None of these points are equilibria for player 2.
If \(p=1\), then \((2(1(pq)^2)1)\) must be nonnegative, so \(q\) must be at least \(11/\sqrt{2}\). This is in equilibrium if \(q=1/2\).
If \(q=0\), then \(p\) must be at least \(1/2\), This in in equilibrium if \(p=1/\sqrt{2}\).
If q=1, then \(p\) must be at least \(1/2\), and there is no equilibrium.
Thus, there are three entangled equilibria (1,1), (1,1/2), (1/\(\sqrt{2}\),0).
