Logit Dynamics with Concurrent Updates for Local Interaction Potential Games

Abstract

Logit choice dynamics constitute a family of randomized best response dynamics based on the logit choice function (McFadden in Frontiers in econometrics. Academic Press, New York, 1974) that models players with limited rationality and knowledge. In this paper we study the all-logit dynamics [also known as simultaneous learning (Alós-Ferrer and Netzer in Games Econ Behav 68(2):413–427, 2010)], where at each time step all players concurrently update their strategies according to the logit choice function. In the well studied (one-)logit dynamics (Blume in Games Econ Behav 5(3):387–424, 1993) instead at each step only one randomly chosen player is allowed to update. We study properties of the all-logit dynamics in the context of local interaction potential games, a class of games that has been used to model complex social phenomena (Montanari and Saberi 2009; Peyton in The economy as a complex evolving system. Oxford University Press, Oxford, 2003) and physical systems (Levin et al. in Probab Theory Relat Fields 146(1–2):223–265, 2010; Martinelli in Lectures on probability theory and statistics. Springer, Berlin, 1999). In a local interaction potential game players are the vertices of a social graph whose edges are two-player potential games. Each player picks one strategy to be played for all the games she is involved in and the payoff of the player is the sum of the payoffs from each of the games. We prove that local interaction potential games characterize the class of games for which the all-logit dynamics is reversible. We then compare the stationary behavior of one-logit and all-logit dynamics. Specifically, we look at the expected value of a notable class of observables, that we call decomposable observables. We prove that the difference between the expected values of the observables at stationarity for the two dynamics depends only on the rationality level $\beta$ and on the distance of the social graph from a bipartite graph. In particular, if the social graph is bipartite then decomposable observables have the same expected value. Finally, we show that the mixing time of the all-logit dynamics has the same twofold behavior that has been highlighted in the case of the one-logit: for some games it exponentially depends on the rationality level $\beta$, whereas for other games it can be upper bounded by a function independent from $\beta$.

Appendix: Mixing Time of the All-Logit for the Curie–Weiss Model

Here we prove upper and lower bounds on the mixing time of the all-logit dynamics for a special graphical coordination game, the CW-game. In such a game we set $a = b = +1$ and $c = d = -1$. Thus, the utility of player $i \in [n]$ is the sum of the number of players playing the same strategy as $i$, minus the number of players playing the opposite strategy; that is, the utility of player $i \in [n]$ at profile ${\mathbf {x}}= (x_1, \ldots , x_n) \in \{-1,+1\}^n$ is

$$\begin{aligned} u_i({\mathbf {x}}) = x_i \sum _{j\ne i} x_j. \end{aligned}$$

It is easy to see that the potential function for this game is

$$\begin{aligned} \Phi ({\mathbf {x}}) = - \sum _{\{i,j\} \in \left( {\begin{array}{c}[n]\\ 2\end{array}}\right) } x_i x_j. \end{aligned}$$

Due to the high level of symmetry of the game, the potential of a profile ${\mathbf {x}}$ depends only on the number of players playing $\pm 1$. Indeed, we can rewrite the potential of ${\mathbf {x}}$ as

$$\begin{aligned} \Phi ({\mathbf {x}}) = - \frac{{\mathsf {Diff}}^2({\mathbf {x}}) - n}{2}, \end{aligned}$$

where ${\mathsf {Diff}}$ is the observable described in Sect. 5.1.

The upper bound Observe that, for the Curie–Weiss model we have $\Delta U = 2n(n-1)$, hence by using Theorem 6.3 we get directly that

$$\begin{aligned} {t_{\mathrm{mix}}}= {\mathcal {O}}\left( e^{2 \beta n (n-1)} \right) . \end{aligned}$$

(16)

Hence it follows that mixing time is ${\mathcal {O}}(1)$ for $\beta = {\mathcal {O}}(1/n^2)$ and it is ${\mathcal {O}}(\text {poly}(n))$ for $\beta = {\mathcal {O}}(\log n / n^2)$.

In what follows we show that factor “2” at the exponent in (16) can be removed and that a slightly better upper bound can be given for $\beta > \log n / n$.

Lemma 7.1

For every ${\mathbf {x}},{\mathbf {y}}\in \Omega$ it holds that

$$\begin{aligned} P({\mathbf {x}},{\mathbf {y}}) \geqslant q^{(n+|{\mathsf {Diff}}({\mathbf {y}})|)/2} (1-q)^{(n-|{\mathsf {Diff}}({\mathbf {y}})|)/2} \end{aligned}$$

where

$$\begin{aligned} q= \frac{1}{1 + e^{2 \beta (n-1)}}. \end{aligned}$$

Proof

Consider a profile ${\mathbf {y}}\in \{-1,+1\}^n$. Observe that the number of players playing $+1$ and $-1$ in ${\mathbf {y}}$ can be written as $\frac{n + {\mathsf {Diff}}({\mathbf {y}})}{2}$ and $\frac{n - {\mathsf {Diff}}({\mathbf {y}})}{2}$, respectively. If ${\mathsf {Diff}}({\mathbf {y}}) > 0$, i.e. if the number of players playing $+1$ is larger than the number of players playing $-1$, then the profile that minimizes $P({\mathbf {x}},{\mathbf {y}})$ is profile ${\mathbf {x}}_{-} = (-1, \ldots , -1)$ where every player plays $-1$. If we name

$$\begin{aligned} q = \frac{e^{- \beta (n-1)}}{e^{- \beta (n-1)} + e^{\beta (n-1)}} = \frac{1}{1+e^{2\beta (n-1)}} \end{aligned}$$

the probability that a player in ${\mathbf {x}}_{-}$ chooses strategy $+1$ for the next round, we have that

$$\begin{aligned} P({\mathbf {x}}_{-},{\mathbf {y}}) = q^{\frac{n + {\mathsf {Diff}}({\mathbf {y}})}{2}} (1-q)^{\frac{n - {\mathsf {Diff}}({\mathbf {y}})}{2}}. \end{aligned}$$

On the other hand, if ${\mathsf {Diff}}({\mathbf {y}}) < 0$, then $P({\mathbf {x}},{\mathbf {y}})$ is minimized when ${\mathbf {x}}= {\mathbf {x}}_{+} = (+1,\ldots ,+1)$ and, since $q$ is also the probability that a player in ${\mathbf {x}}_{+}$ chooses strategy $-1$ for the next round, we have that

$$\begin{aligned} P({\mathbf {x}}_{+},{\mathbf {y}}) = q^{\frac{n - {\mathsf {Diff}}({\mathbf {y}})}{2}} (1-q)^{\frac{n + {\mathsf {Diff}}({\mathbf {y}})}{2}} \end{aligned}$$

and the thesis follows. $\square$

Now we can give an upper bound on the mixing time by using Lemmata 6.1 and 7.1.

Theorem 7.2

(Upper bound) The mixing time of the all-logit dynamics for the Curie–Weiss model is

$$\begin{aligned} {t_{\mathrm{mix}}}= {\mathcal {O}}\left( n e^{\beta n^2} \right) . \end{aligned}$$

If $\beta \geqslant \log n / n$ the mixing time is

$$\begin{aligned} {t_{\mathrm{mix}}}= {\mathcal {O}}\left( \frac{n e^{\beta n^2}}{2^n} \right) . \end{aligned}$$

Proof

From Lemma 7.1 it follows that for every ${\mathbf {y}}\in \{-1,+1 \}^n$ we have

$$\begin{aligned} \alpha _{{\mathbf {y}}} = \min \{ P({\mathbf {x}},{\mathbf {y}}) \mid {\mathbf {x}}\in \{-1,+1\}^n \} \geqslant q^{(n+|{\mathsf {Diff}}({\mathbf {y}})|)/2} (1-q)^{(n-|{\mathsf {Diff}}({\mathbf {y}})|)/2}. \end{aligned}$$

Hence

$$\begin{aligned} \alpha = \sum _{{\mathbf {y}}\in \{-1,+1\}^n} \alpha _{{\mathbf {y}}} \geqslant \sum _{{\mathbf {y}}\in \{-1,+1\}^n} q^{(n+|{\mathsf {Diff}}({\mathbf {y}})|)/2} (1-q)^{(n-|{\mathsf {Diff}}({\mathbf {y}})|)/2}. \end{aligned}$$

(17)

Now observe that there are $\left( {\begin{array}{c}n\\ \frac{n-k}{2}\end{array}}\right)$ profiles ${\mathbf {y}}$ such that ${\mathsf {Diff}}({\mathbf {y}}) = k$, and since $q \leqslant 1/2$, the largest terms in (17) are the ones such that ${\mathsf {Diff}}({\mathbf {y}})$ is as close to zero as possible. In order to give a lower bound to $\alpha$ we will thus consider only profiles ${\mathbf {y}}$ such that ${\mathsf {Diff}}({\mathbf {y}}) = 0$, when $n$ is even, and profiles ${\mathbf {y}}$ such that ${\mathsf {Diff}}({\mathbf {y}}) = \pm 1$, when $n$ is odd.

Case $n$ even: If we consider only profiles ${\mathbf {y}}$ such that ${\mathsf {Diff}}({\mathbf {y}}) = 0$ in (17) we have that

$$\begin{aligned} \alpha \geqslant \left( {\begin{array}{c}n\\ n/2\end{array}}\right) [q(1-q)]^{n/2}. \end{aligned}$$

By using a standard lower bound for the binomial coefficient (see e.g. Lemma 9.2 in [36]) we have that

$$\begin{aligned} \left( {\begin{array}{c}n\\ n/2\end{array}}\right) \geqslant \frac{2^n}{n+1}. \end{aligned}$$

As for $[q(1-q)]^{n/2}$ we have that

$$\begin{aligned} q(1-q)&= \frac{1}{1+e^{2\beta (n-1)}} \cdot \frac{1}{1+e^{-2\beta (n-1)}} \nonumber \\&= \frac{1}{e^{2\beta (n-1)} + 2 + e^{-2\beta (n-1)}} \nonumber \\&= \frac{1}{e^{2\beta (n-1)} \left( 1 + 2 e^{-2\beta (n-1)} + e^{-4 \beta (n-1)} \right) }. \end{aligned}$$

(18)

Now observe that for every $\beta \geqslant 0$ we can bound $1 + 2 e^{-2\beta (n-1)} + e^{-4 \beta (n-1)} \leqslant 4$. Thus we have that

$$\begin{aligned}{}[q(1-q)]^{n/2} \geqslant \frac{1}{2^n e^{\beta n(n-1)}}. \end{aligned}$$

(19)

Hence

$$\begin{aligned} \alpha \geqslant \left( {\begin{array}{c}n\\ n/2\end{array}}\right) [q(1-q)]^{n/2} \geqslant \frac{1}{(n+1) e^{\beta n(n-1)}}. \end{aligned}$$

And by using Lemma 6.1 we have

$$\begin{aligned} {t_{\mathrm{mix}}}= {\mathcal {O}}\left( n e^{\beta n(n-1)} \right) . \end{aligned}$$

If $\beta$ is large enough, say $\beta \geqslant \log n / n$, in (18) we can bound

$$\begin{aligned} 1 + 2 e^{-2\beta (n-1)} + e^{-4 \beta (n-1)} \leqslant 1 + \frac{1}{n}. \end{aligned}$$

Thus, in this case we have that

$$\begin{aligned}{}[q(1-q)]^{n/2} \geqslant \frac{1}{e^{\beta n (n-1)} \left( 1 + 1/n \right) ^{(n/2)}} \geqslant \frac{1}{e^{\beta n(n-1)} \cdot \sqrt{e}}. \end{aligned}$$

(20)

Hence $\alpha \geqslant \frac{2^n}{(n+1) e^{1/2 + \beta n(n-1)}}$ and

$$\begin{aligned} {t_{\mathrm{mix}}}= {\mathcal {O}}\left( \frac{n e^{\beta n(n-1)}}{2^n} \right) . \end{aligned}$$

Case $n$ odd: If we consider only profiles ${\mathbf {y}}$ such that ${\mathsf {Diff}}({\mathbf {y}}) = \pm 1$ in (17) we get

$$\begin{aligned} \alpha \geqslant 2 \left( {\begin{array}{c}n\\ \frac{n+1}{2}\end{array}}\right) q^{\frac{n+1}{2}} (1-q)^{\frac{n-1}{2}} = 2 \left( {\begin{array}{c}n\\ \frac{n+1}{2}\end{array}}\right) \left( q(1-q)\right) ^{n/2} \sqrt{\frac{q}{1-q}}. \end{aligned}$$

Now observe that

$$\begin{aligned} \sqrt{\frac{q}{1-q}} = e^{-\beta (n-1)} \qquad \text{ and } \qquad \left( {\begin{array}{c}n\\ \frac{n+1}{2}\end{array}}\right) \geqslant \frac{1}{2} \cdot \frac{2^n}{n+1}. \end{aligned}$$

By using bounds (19) and (20) for $[q(1-q)]^{n/2}$ we get ${t_{\mathrm{mix}}}= {\mathcal {O}}\left( n e^{\beta (n^2 - 1)} \right)$ for every $\beta \geqslant 0$ and ${t_{\mathrm{mix}}}= {\mathcal {O}}\left( \frac{n e^{\beta (n^2-1)}}{2^n} \right)$ for $\beta \geqslant \log n / n$. $\square$

The lower bound In order to give a lower bound on the mixing time, we first show that, for the Curie–Weiss model, $K({\mathbf {x}},{\mathbf {y}})$ can be written as a function of ${\mathsf {Diff}}({\mathbf {x}})$, ${\mathsf {Diff}}({\mathbf {y}})$ and of the Hamming distance between the two profiles.

Lemma 7.3

Let ${\mathbf {x}},{\mathbf {y}}\in \{-1,+1\}^n$ be two profiles with magnetization ${\mathsf {Diff}}({\mathbf {x}})$ and ${\mathsf {Diff}}({\mathbf {y}})$ respectively and let $h_{{\mathbf {x}},{\mathbf {y}}}$ be their Hamming distance, i.e. the number of players where they differ. Then

$$\begin{aligned} K({\mathbf {x}}, {\mathbf {y}}) = n - {\mathsf {Diff}}({\mathbf {x}}) \cdot {\mathsf {Diff}}({\mathbf {y}}) - 2 h_{{\mathbf {x}},{\mathbf {y}}}. \end{aligned}$$

Proof

As stated above, $\Phi ({\mathbf {x}}) = \frac{n - {\mathsf {Diff}}^2({\mathbf {x}})}{2}$. In order to evaluate $K({\mathbf {x}},{\mathbf {y}}) = \sum _{i=1}^n \Phi ({\mathbf {x}}_{-i}, y_i) - (n - 2) \Phi ({\mathbf {x}})$ let us name $n_1,n_2$ and $n_3$ as follows

$$\begin{aligned} n_1&= \# \{ i \in [n] :x_i = y_i \};\\ n_2&= \# \{ i \in [n] :x_i = +1, y_i = -1 \};\\ n_3&= \# \{ i \in [n] :x_i = -1, y_i = +1 \}. \end{aligned}$$

In other words, $n_1$ is the number of players playing the same strategy in profiles ${\mathbf {x}}$ and ${\mathbf {y}}$, $n_2$ is the number of players playing $+1$ in ${\mathbf {x}}$ and $-1$ in ${\mathbf {y}}$, and $n_3$ the number of players playing $-1$ in ${\mathbf {x}}$ and $+1$ in ${\mathbf {y}}$. It holds that

$$\begin{aligned} \sum _{i=1}^n \Phi ({\mathbf {x}}_{-i}, y_i)&= n_1 \frac{n- {\mathsf {Diff}}^2({\mathbf {x}})}{2} + n_2 \frac{n - ({\mathsf {Diff}}({\mathbf {x}}) -2)^2}{2} + n_3 \frac{n - ({\mathsf {Diff}}({\mathbf {x}})+2)^2}{2}\nonumber \\&= \frac{1}{2} \Big ( (n_1+n_2+n_3) (n - {\mathsf {Diff}}^2({\mathbf {x}}))\nonumber \\&+\, 4(n_2-n_3) {\mathsf {Diff}}({\mathbf {x}}) - 4 (n_2+n_3) \Big ). \end{aligned}$$

(21)

Now observe that $n_1+n_2+n_3 = n$, $2(n_2-n_3) = {\mathsf {Diff}}({\mathbf {x}}) - {\mathsf {Diff}}({\mathbf {y}})$, and $(n_2+n_3) = h_{{\mathbf {x}},{\mathbf {y}}}$. Hence from (21) we get

$$\begin{aligned} \begin{aligned} \sum _{i=1}^n \Phi ({\mathbf {x}}_{-i}, y_i)&= \frac{1}{2} \left( n (n+{\mathsf {Diff}}^2({\mathbf {x}})) + 2({\mathsf {Diff}}({\mathbf {x}})-{\mathsf {Diff}}({\mathbf {y}})) {\mathsf {Diff}}({\mathbf {x}}) -4 h_{{\mathbf {x}},{\mathbf {y}}}\right) \\&= \frac{n^2}{2} - \frac{n-2}{2} {\mathsf {Diff}}^2({\mathbf {x}}) - {\mathsf {Diff}}({\mathbf {x}}){\mathsf {Diff}}({\mathbf {y}}) - 2 h_{{\mathbf {x}},{\mathbf {y}}}. \end{aligned} \end{aligned}$$

(22)

Thus $K({\mathbf {x}},{\mathbf {y}}) = n - {\mathsf {Diff}}({\mathbf {x}}) \cdot {\mathsf {Diff}}({\mathbf {y}}) - 2 h_{{\mathbf {x}},{\mathbf {y}}}$. $\square$

Since the Hamming distance between two profiles is at most $n$, from the above lemma we get the following observation.

Observation 7.4

Let ${\mathbf {x}},{\mathbf {y}}$ be two profiles with ${\mathsf {Diff}}({\mathbf {x}}) \cdot {\mathsf {Diff}}({\mathbf {y}}) \leqslant 0$, then $K({\mathbf {x}},{\mathbf {y}}) \geqslant -n$.

Now we can give a lower bound on the mixing time by using the bottleneck-ratio technique.

Theorem 7.5

(Lower bound) The mixing time of the all-logit dynamics for the Curie–Weiss model is

$$\begin{aligned} {t_{\mathrm{mix}}}= \Omega \left( \frac{e^{\beta n(n - 2)}}{4^n} \right) . \end{aligned}$$

Proof

Let $S_- \subseteq \{-1,+1\}^n$ be the set of profiles ${\mathbf {x}}$ such that ${\mathsf {Diff}}({\mathbf {x}}) < 0$, i.e.

$$\begin{aligned} S_- = \{ {\mathbf {x}}\in \{-1,+1\}^n :{\mathsf {Diff}}({\mathbf {x}}) < 0 \} \end{aligned}$$

and observe that $\pi (S_-) \leqslant 1/2$. From Observation 7.4 we have that for every ${\mathbf {x}}\in S_-$ and ${\mathbf {y}}\in S_+ = \{-1,+1\}^n \setminus S_-$ it holds that

$$\begin{aligned} \pi ({\mathbf {x}})P({\mathbf {x}},{\mathbf {y}}) = \frac{e^{-\beta K({\mathbf {x}},{\mathbf {y}})}}{Z} \leqslant \frac{e^{\beta n}}{Z}. \end{aligned}$$

(23)

Moreover, if we name ${\mathbf {x}}_{-}$ the profile where everyone is playing $-1$ we have that

$$\begin{aligned} \pi (S_-) \geqslant \pi ({\mathbf {x}}_{-}) \geqslant \frac{1}{Z} e^{-2\beta \Phi ({\mathbf {x}}_{-})} = \frac{1}{Z} e^{\beta n(n-1)}. \end{aligned}$$

(24)

Hence, by using bounds (23) and (24), and the fact that the size of $S_-$ is at most $2^{n-1}$, we can bound the bottleneck at $S_-$ with

$$\begin{aligned} B(S_-) = \frac{Q(S_-,S_+)}{\pi (S_-)} = \frac{\sum _{{\mathbf {x}}\in S_-} \sum _{{\mathbf {y}}\in S_+} \pi ({\mathbf {x}}) P({\mathbf {x}},{\mathbf {y}})}{\pi (S_-)} \leqslant \frac{2^{2n-2} e^{\beta n}}{e^{\beta n(n-1)}} = \frac{2^{2n-2}}{e^{\beta n (n-2)}}. \end{aligned}$$

By using the bottleneck-ratio theorem (see e.g. Theorem 7.3 in [32]) it follows that

$$\begin{aligned} {t_{\mathrm{mix}}}= \Omega \left( \frac{e^{\beta n (n-2)}}{2^{2n}} \right) . \end{aligned}$$

$\square$

Remark

In this section we proved upper and lower bounds on the mixing time of the all-logit dynamics for the Curie–Weiss model. In particular, the upper bound shows that for $\beta = {\mathcal {O}}(1/n^2)$ the mixing time is constant and for $\beta = {\mathcal {O}}( \log n / n^2)$ it is at most polynomial. The lower bound shows that, for every constant $\varepsilon > 0$, if $\beta > (1+\varepsilon ) (\log 4) /n$ the mixing time is exponential. When $\beta$ is between $\Theta (\log n /n^2)$ and $\Theta (1/n)$ we still cannot say if mixing is polynomial or exponential. This is to be compared with the mixing time of the one-logit dynamics: in this case, the dynamics are known to quickly converge to the stationary distribution for $\beta = {\mathcal {O}}(\log n / n^2)$ and to take super-polynomial time for $\beta = \omega (\log n / n^2)$ [6]. Hence, the mixing time of the all-logit asymptotically matches the mixing time of the one-logit.