Frustrated Equilibrium of Asymmetric Coordinating Dynamics in a Marketing Game

Abstract

This paper considers a recently introduced model for socially-contingent decision-making and addresses the connection between influences on individual decision-making and the statistical, information-theoretic properties associated with such decision-making dynamics on a social network. In particular, we analytically show, on a few simple examples, the correspondence between coordinating influences and positively correlated models which in turn correspond to models with entropy that decreases monotonically in the strength of the influences. Moreover, we discuss numerical results that suggest asymmetric yet coordinating influences may converge to a frustrated equilibrium.

M. G. Reyes—Independent Researcher and Consultant.

A Appendix: Sketches of Derivations

Note that to determine whether a particular covariance is positive or negative, one needs to determine the sign of the numerator, which we creatively refer to as numerator. For both the chain and the cycle, numerator will be a product of factors, one of which will be positive or negative depending on whether the direct biases agree in polarity, another of which will be positive or negative depending on whether there are an odd or an even number of negative edges between the two direct biases.

Each factor in numerator will be a sum of terms, each of term of which itself a product $\sinh {\uptheta }$’s and $\cosh {\uptheta }$’s, where ${\uptheta }$ is a direct or social bias. Without loss of generality, we simplify analysis by assuming uniform social biases, i.e., ${\uptheta }_{i,i+1} = {\uptheta }$. For both a chain and a cycle, direct biases are zero, except for two sites. In the chain, the nonzero direct biases are at sites 1 and N, the ends of the chain. In the cycle, the nonzero direct biases are at sites 1 and $k < N$. In the interest of space, we will economize notation by letting $C_{{\uptheta }} \,{{\mathop {=}\limits ^{\varDelta }}}\,\cosh {\uptheta }$ and $S_{{\uptheta }} \,{{\mathop {=}\limits ^{\varDelta }}}\,\sinh {\uptheta }$ when ${\uptheta }$ is a social bias, and $C_i \,{{\mathop {=}\limits ^{\varDelta }}}\,\cosh {\uptheta }_i$ and $S_i \,{{\mathop {=}\limits ^{\varDelta }}}\,\sinh {\uptheta }_i$ when ${\uptheta }_i$ is a nonzero direct bias at site i.

1.1 A.1 Theorems 1 and 2

Proof

(Theorem 1). Using the concept of the transfer matrix [3], one can compute the partition function $Z({\varvec{{\uptheta }}})$ as

$$\begin{aligned} Z({\varvec{{\uptheta }}})&= \left[ \begin{array}{cc} e^{{\uptheta }_{N}} &{} e^{-{\uptheta }_{N}} \\ \end{array}\right] \left[ \begin{array}{cc} 1 &{} 1 \\ 1 &{} -1 \\ \end{array}\right] \left[ \begin{array}{cc} \overline{C}_{{\uptheta }}^{N-1} &{} 0 \\ 0 &{} \overline{S}_{{\uptheta }}^{N-1} \\ \end{array}\right] \left[ \begin{array}{cc} 1 &{} 1 \\ 1 &{} -1 \\ \end{array}\right] \left[ \begin{array}{c} e^{{\uptheta }_{1}} \\ e^{-{\uptheta }_{1}} \\ \end{array}\right] \\&= \overline{C}_{{\uptheta }_{N}} \overline{C}_{{\uptheta }}^{N-1} \overline{C}_{{\uptheta }_{1}} + \overline{S}_{{\uptheta }_{N}} \overline{S}_{{\uptheta }}^{N-1} \overline{S}_{{\uptheta }_{1}} . \end{aligned}$$

   $\square$

We now sketch a proof that ${\varvec{t}}$ is positively correlated if and only if ${\varvec{t}}$ is non-frustrated.

Proof

(Theorem 2). We will determine the sign of numerator for the covariance $\mathbf{{cov}}\left( t_1,t_{12}\right)$ between statistics $t_1(x_1)$ and $t_{12}(x_1,x_2)$. In the case that ${\uptheta }_1 > 0$ and ${\uptheta }_N > 0$, it is straightforward to show

$$\begin{aligned} \mathbf{{numerator}}&= C_N S_N \left[ C_{{\uptheta }}^{N-1} C_{{\uptheta }} S_{{\uptheta }}^{N-2} - S_{{\uptheta }} C_{{\uptheta }}^{N-2} S_{{\uptheta }}^{N-1}\right] \left[ C_1^2 - S_1^2\right] \end{aligned}$$

(13)

The first factor $C_N S_N$ is positive since ${\uptheta }_N > 0$. The third factor, $C_1^2 - S_1^2$, is positive for all ${\uptheta }_1$.

Consider the middle factor, $C_{{\uptheta }}^{N-1} C_{{\uptheta }} S_{{\uptheta }}^{N-2} - S_{{\uptheta }} C_{{\uptheta }}^{N-2} S_{{\uptheta }}^{N-1}$. There are four cases to consider. First, whether there are an odd or even number of negative edges in the chain. For each of these cases, whether the social bias ${\uptheta }_{12}$ is positive or negative. One can see that when there are an odd number of negative edges on the chain

$$\begin{aligned} C_{{\uptheta }}^{N-1} C_{{\uptheta }} S_{{\uptheta }}^{N-2} - S_{{\uptheta }} C_{{\uptheta }}^{N-2} S_{{\uptheta }}^{N-1}&< 0 , \end{aligned}$$

and when there are an even number of negative edges along the chain,

$$\begin{aligned} C_{{\uptheta }}^{N-1} C_{{\uptheta }} S_{{\uptheta }}^{N-2} - S_{{\uptheta }} C_{{\uptheta }}^{N-2} S_{{\uptheta }}^{N-1}&> 0 . \end{aligned}$$

Combining this with (13), we see that $\mathbf{{cov}}\left( t_1,t_{12}\right) < 0$ when ${\varvec{t}}$ is frustrated and $\mathbf{{cov}}\left( t_1,t_{12}\right) > 0$ is non-frustrated. Note, however, that in order to show positive correlation, one has to show that all covariances are positive when ${\varvec{t}}$ is non-frustrated. By recalling (13), one can convince themselves that this is the case.

The case where ${\uptheta }_1 < 0$ and ${\uptheta }_N > 0$ follows similarly, though one should recall (10) and (11) for sign reversals.    $\square$

1.2 A.2 Proof Sketch for Theorem 4

The concept of the transfer matrix can again be used to compute the partition function $Z({\varvec{{\uptheta }}})$ in the case of an Ising model on a cycle. However, one does so by conditioning on a particular site, say $i_0$, and computing the respective conditional partition functions $Z_{i_0}^{(A)}$ and $Z_{i_0}^{(B)}$ corresponding to conditioning that $x_{i_0} = 1$ and $X_{i_0} = -1$. Then, $Z = Z_{i_0}^{(A)} + Z_{i_0}^{(B)}$. We begin with the following lemma.

Lemma 1

Consider a cycle with M direct biases, at sites $i_1,\dots ,i_M$. The conditional partition function $Z_{i_0}^{(A)}$ and $Z_{i_0}^{(B)}$ are, respectively,

$$\begin{aligned}&Z_{i_0}^{(A)} \\&= H^0 \prod \limits _{k=1}^M \overline{C}_k + \sum \limits _{j=1}^{M-1} H^{i_j,i_{j+1}} \overline{S}_{i_j} \overline{S}_{i_{j+1}} \!\!\!\!\!\! \prod \limits _{k\not =j,j+1} \!\!\!\!\! \overline{C}_k + \prod \limits _{k\not =1} \overline{C}_k \overline{S}_1 H^{i_0,i_1} + \prod \limits _{k\not =M} \!\! \overline{C}_k \overline{S}_M H^{i_0,i_M} , \nonumber \\&{\textit{and}} \nonumber \\&Z_{i_0}^{(B)} \nonumber \\&= H^0 \prod \limits _{k=1}^M \overline{C}_k + \sum \limits _{j=1}^{M-1} H^{i_j,i_{j+1}} \overline{S}_{i_j} \overline{S}_{i_{j+1}} \!\!\!\!\!\! \prod \limits _{k\not =j,j+1} \!\!\!\!\! \overline{C}_k - \prod \limits _{k\not =1} \overline{C}_k \overline{S}_1 H^{i_0,i_1} - \prod \limits _{k\not =M} \!\! \overline{C}_k \overline{S}_M H^{i_0,i_M} . \nonumber \end{aligned}$$

(14)

Proof

We will prove (14) by induction. It is straightforward¹ to verify the conclusion of the lemma in the case of three direct biases, which establishes the base step. That is, the statement holds for some $M\le N$. For the inductive step, we will first compute $Z_{i_0}^{(A)}$ as a function of the message $\overline{m}$ from site $i_M$ to site $i_M + 1$, and establish a relationship between $\overline{m}$ and $Z_{i_0}^{(A)}$ in the base case. We will then compute $Z_{i_0}^{(A)}$ as a function of $\overline{m}$ in the inductive case of $M + 1$ direct biases. For simplicity and without loss of generality, assume that for all $\{i,i+1\}$, ${\uptheta }_{ij} = {\uptheta }$.

Suppose there are M direct biases. We compute $Z_{i_0}^{(A)}$ by conditioning on site $i_0 = 0$. The sites where there exist direct biases are enumerated as $i_1,\ldots ,i_M$. Moreover, let $\varDelta$ be the distance from $i_M$ to $i_0$ in the ‘forward’ direction, i.e., the direction in which no other direct biases exist between $i_M$ and $i_0$. One can show that

$$\begin{aligned} Z_{i_0}^{(A)}&= C_{{\uptheta }}^{\varDelta }\left[ \overline{m}(A) + \overline{m}(B) \right] + S_{{\uptheta }}^{\varDelta } \left[ \overline{m}(A) - \overline{m}(B) \right] . \end{aligned}$$

Equating this with (14), which is established for the case of M direct biases, one can see that

$$\begin{aligned} \overline{m}(A) + \overline{m}(B)&= \alpha _0 C_{{\uptheta }}^{N-\varDelta } + \sum \limits _{j=1}^{M-1} \alpha _{i_j,i_{j+1}} C_{{\uptheta }}^{i_j - i_{j+1} - \varDelta } S_{{\uptheta }}^{i_{j+1} - i_j} \nonumber \\&\,\,\,\, + \alpha _{0,1} C_{{\uptheta }}^{i_0 - i_1 - \varDelta } S_{{\uptheta }}^{i_1 - i_0} + \alpha _{M,0} C_{{\uptheta }}^{i_M - i_0 - \varDelta } S_{{\uptheta }}^{i_0 - i_M}\!, \end{aligned}$$

(15)

$$\begin{aligned} \mathrm{{and}} \nonumber \\ \overline{m}(A) - \overline{m}(B)&= \alpha _0 S_{{\uptheta }}^{N-\varDelta } + \sum \limits _{j=1}^{M-1} \alpha _{i_j,i_{j+1}} C_{{\uptheta }}^{i_{j+1} - i_{j}} S_{{\uptheta }}^{i_{j} - i_{j+1} - \varDelta } \nonumber \\&\,\,\,\, + \alpha _{0,1} C_{{\uptheta }}^{i_1 - i_0} S_{{\uptheta }}^{i_0 - i_1 - \varDelta } + \alpha _{M,0} C_{{\uptheta }}^{i_0 - i_M} S_{{\uptheta }}^{i_M - i_0 - \varDelta } . \end{aligned}$$

(16)

Now consider the inductive case of $M+1$ direct biases, where $i_1\ldots ,i_M$ are as they were in the base case, and $i_{M+1} > i_M$, i.e., it is the only direct bias between $i_M$ and $i_0$. Note that $\overline{m}$ is the same as in the base case of M direct biases, and therefore hence $\overline{m}(A) - \overline{m}(B)$ and $\overline{m}(A) + \overline{m}(B)$ are the same as well. One can show that

$$\begin{aligned} Z_{i_0}(A)&= \left( C_M C_{{\uptheta }}^{\varDelta } + S_m C_{{\uptheta }}^{\varDelta _1} S_{{\uptheta }}^{\varDelta _2} \right) \left[ \overline{m}(A) + \overline{m}(B) \right] \\&\,\,\,\,\,\,\,\, + \left( C_M S_{{\uptheta }}^{\varDelta } + S_m S_{{\uptheta }}^{\varDelta _1} C_{{\uptheta }}^{\varDelta _2} \right) \left[ \overline{m}(A) - \overline{m}(B) \right] \end{aligned}$$

Substituting (15) and (16) establishes (14) for the inductive case of $M+1$ direct biases.    $\square$

We now sketch a proof of Theorem 4.

Proof

(Theorem 4). Let ${\uptheta }_1$ and ${\uptheta }_2$ denote the non-zero direct biases on the cycle. We assume that ${\uptheta }_2 > 0$. We will consider four cases. First, whether ${\uptheta }_1 < 0$ or ${\uptheta }_1 > 0$. Second, whether there are an odd or even number of anti-coordinating social biases along each path between $i_1$ and $i_2$.

In the case that ${\uptheta }_1 < 0$,

$$\begin{aligned} \mathbf{{numerator}}&= H^0 H^{1,2} \left[ C_1^2 S_2^2 + S_1^2 S_2^2 - S_1^2 C_2^2 - C_1^2 C_2^2 \right] \end{aligned}$$

(17)

where it is straightforward to verify that

$$\begin{aligned} C_1^2 S_2^2 + S_1^2 S_2^2 - S_1^2 C_2^2 - C_1^2 C_2^2&< 0 . \end{aligned}$$

If there are an odd number of anti-coordinating social biases along each path connecting $i_1$ and $i_2$,

$$\begin{aligned} H^0 H^{1,2}&= \left[ C_{{\uptheta }}^N + S_{{\uptheta }}^N \right] \left[ - C_{{\uptheta }}^{\varDelta _1} S_{{\uptheta }}^{\varDelta _2} - C_{{\uptheta }}^{\varDelta _2} S_{{\uptheta }}^{\varDelta _1} \right] < 0 , \end{aligned}$$

(18)

while if there are an even number of anti-coordinating social biases along each path connecting $i_1$ and $i_2$, we have

$$\begin{aligned} H^0 H^{1,2}&= \left[ C_{{\uptheta }}^N + S_{{\uptheta }}^N \right] \left[ C_{{\uptheta }}^{\varDelta _1} S_{{\uptheta }}^{\varDelta _2} + C_{{\uptheta }}^{\varDelta _2} S_{{\uptheta }}^{\varDelta _1} \right] > 0 . \end{aligned}$$

(19)

Substituting (18) into (17) shows that when the two direct biases are of opposite polarity, and there are an odd number of anti-coordinating social along each path connecting $i_1$ and $i_2$, then the covariance between $t_1$ and $t_2$ is positive. Substituting (19) into (17) shows that when the two direct biases oppose in polarity, and there are an even number of anti-coordinating social biases along each path connecting $i_1$ and $i_2$, then the covariance between $t_1$ and $t_2$ is negative.

In the case that ${\uptheta }_1 > 0$,

$$\begin{aligned} \mathbf{{numerator}}&= H^0 H^{1,2} \left[ C_1^2 C_2^2 + S_1^2 S_2^2 - S_1^2 C_2^2 - C_1^2 S_2^2 \right] , \end{aligned}$$

(20)

where one can verify that

$$\begin{aligned} C_1^2 C_2^2 + S_1^2 S_2^2 - S_1^2 C_2^2 - C_1^2 S_2^2&> 0 . \end{aligned}$$

Substituting (18) and (19) into (20) will yield analogous results.    $\square$