Global Games with Noisy Information Sharing

# Global Games with Noisy Information Sharing

Hessam Mahdavifar, Ahmad Beirami, Behrouz Touri, and Jeff S. Shamma The work of B. Touri is supported by National Science Foundation under the grant NSF-ECCS 1610003. The work of J. S. Shamma is supported by funding from King Abdullah University of Science and Technology (KAUST).H. Mahdavifar is with the Department of Electrical and Computer Engineering, University of Michigan, Ann Arbor, MI 48109, USA (email: hessam@umich.edu).A. Beirami is with the School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA, and also with the Electrical Engineering and Computer Science Department, Massachusetts Institute of Technology, Cambridge, MA, USA (email: beirami@seas.harvard.edu, beirami@mit.edu).B. Touri is with University of California San Diego, Electrical and Computer Engineering Department, La Jolla 92093, USA (email: btouri@eng.ucsd.edu).J. S. Shamma is with King Abdullah University of Science and Technology (KAUST), Computer, Electrical and Mathematical Science and Engineering Division (CEMSE), Thuwal 23955–6900, Saudi Arabia (email: jeff.shamma@kaust.edu.sa).
###### Abstract

Global games form a subclass of games with incomplete information where a set of agents decide actions against a regime with an underlying fundamental representing its power. Each agent has access to an independent noisy observation of . In order to capture the behavior of agents in a social network of information exchange we assume that agents share their observation in a noisy environment prior to making their decision. We show that global games with noisy sharing of information do not admit an intuitive type of threshold policy which only depends on agents’ belief about the underlying . This is in contrast to the existing results on the threshold policy for the conventional set-up of global games. Motivated by this result, we investigate the existence of equilibrium strategies in a more general collection of threshold-type policies and show that such equilibrium strategies exist and are unique if the sharing of information happens over a sufficiently noisy environment.

{keywords}

Global game, threshold policy

## I Introduction

Games of incomplete information are central to modeling of socio-economic behaviors in social networks. In games with incomplete information, the information that is shared among the agents is not symmetric and often each agent has access to limited information about the game’s parameters, which may be correlated with the information that is available to the other agents. This incompleteness of information is often represented by considering the payoff as a function of some random variable whose exact value is not known to the agents, which is called the underlying economic or political fundamental in the society.

A subclass of games with incomplete information is the class of global games, originally introduced in [1, 2]. In its simplest form, each agent has access to an independent observation of the underlying fundamental based on which she takes either of the two actions: risky or safe action. The payoff of an agent taking the risky action is monotonically increasing with the number of agents who take the risky action and is a decreasing function of the underlying fundamental.

Global games have been central to modeling and studying many social coordination phenomena. In [1], the authors discussed a general form of global games for two players and discussed how the vanishing noise in a public information results in a unique threshold policy. Such analysis and modeling technique was extended and used in [3] to model currency attacks. The authors model currency attack using a global game formulation and the authors show that even though the complete information variation of the game has multiple (Nash) equilibria, introducing a small noise into speculators observations about the economic fundamental leads to a unique threshold policy equilibrium. In [4], a similar idea is used to model the debt crisis and to arrive at a unique threshold policy for the players given the noisy observations of the players. Finally, in [5], the accuracy of the prediction of the global games results have been extensively studied and the authors concluded that: “Comparing sessions with common and private information, we observe only small differences in behavior.” In the engineering domain, the authors of[6] have discussed the application of global games in distributed task-allocation in collaborative robotic networks and utilized the same idea to show the uniqueness of threshold policies for task allocation in coordination problems in robotics networks.

The majority of the past studies on global games have been focused on the information structure where each agent has an independent noisy observation of the underlying economic or social fundamental, i.e., given the fundamental, the observations of each agent is independent of the rest of the agents’ observations. In such settings, it has been shown that, under some condition on the distribution of the underlying fundamental and the independency of agents’ observations, there exists an equilibrium strategy with threshold policies based on the private observations of each agent. In [7], the case of perfect sharing of information was introduced and the structure of equilibria for global games with perfect sharing of information was discussed. We challenge robustness of these results to sharing of information: we show that even in a simple instance of global games, when agents share information, the intuitive equilibrium doesn’t exist. However, we show that with a proper generalization of the notion of threshold policy, a symmetric threshold policy (in its extended sense), exists under some conditions on the communication noise. This paper extends the results of the authors in [8, 9] and provides an extension of the existence of a symmetric equilibria in global games with noisy sharing of information involving more than two players.

In this paper, global games with noisy sharing of information are introduced where any two agents share their information in a noisy environment. Our contributions are summarized below:

• We show the fragility of the existing results in global games by showing that with noisy sharing of information, an intuitive threshold policy which only depends on agents’ belief about the underlying fundamental does not lead to a Bayesian Nash equilibrium.

• We show that if a certain functional equation has a solution, then a symmetric Bayesian Nash equilibrium exists for global games with noisy sharing of information which can be described as a generalized form of the intuitive threshold policies.

• We utilize Banach fixed point theorem and show that the functional equation has a solution if the noise and the information structure of the underlying model satisfy certain conditions. This establishes the existence of the generalized threshold strategy equilibria, when the noise parameters and the information structure satisfy certain conditions.

It is worth noting that this work also relates to many of the recent attempts for understanding the role of information and information structure in multiagent decision making problems [10, 11, 12, 13]. In this paper, we consider a static framework, similar to most applications of global games. Global games have been also studied with delay [14] and in dynamic framework [15], which also relates to networking games and evolution of cooperation in social networks [16, 17, 18, 19, 20]. The structure of this paper is as follows. In Section II we introduce global games with noisy sharing of information, and we mathematically formulate the problem of interest. In Section III we state the three main theorems of the paper on the non-existence of certain intuitive threshold policies, the existence of a general form of threshold policies, and the convergence to this threshold policy via an iterative method. In Section IV the proofs of the three main theorems are established. Some numerical examples are provided in Section V. Finally, we conclude the paper in Section VI.

## Ii Problem Setup

In this section, we present the framework of the problem that will be studied in this paper. We study the basic form of the global games with noisy sharing of information. In this setting, we consider a set

 [n]:={1,…,n},

of agents or players. Each agent has a set of binary actions . We refer to the action as the risky action and as the safe action. The payoff of an agent taking the safe action is zero, whereas that of taking the risky action is , where is a random variable representing the underlying fundamental in the society. In other words, if is an action profile of the agents, then the utility of the th agent is the function with

 ui(α)=αi(n∑j=1αj−θ). (1)

Remark. For an intuitive description of the utility functions consider global games in the context of political regime change[21]. In this set-up, the parameter represents the power of a political regime that can be overthrown but only if enough citizens participate in an uprising, i.e., take the risky action. It is thus natural to assume that the utility function of agents taking the risky action is increasing in the number of such agents and is decreasing in the fundamental parameter . Also, the utility function of agents taking the safe action is considered to be zero. Therefore, utility functions of the form given in (1) are natural to use in global game models. The results in the global games literature can often be extended to general utility functions that are monotone in the number of agents taking the risky actions as well as .

Observations and Policies: In the standard setting for global games, agent is observing where ’s are identically and independently distributed random variables [1, 2]. In our work, agents share their observations with the other agents in the society through noisy channels. In other words, each agent has its private observation as well as noisy observations of other agents’ private observations. Mathematically, we represent agent ’s overall observation by a random vector (which relates to and other agents’ observations). The parameter represents the number of observations that agent has. In this paper, we focus on the case that each agent shares its observation with all other agents. Hence, , for all . We refer to as the (private) information of agent (about ). We refer to a measurable function that maps a private observation of agent to one of the two actions as a (pure) strategy or policy. When for some , we say that is a threshold policy if for and for , for some threshold value . We denote such a strategy by . In almost all the instances of the global games, the random variables have continuous joint distribution, and hence, the value of the strategy at the threshold value is practically unimportant.

Equilibrium: Our focus in this paper is on the existence of a strategy profile that results in a Bayesian Nash Equilibrium. To introduce this concept, let be a strategy profile of the agents and let be the strategy profile of the agents except the th agent’s strategy. We say that a best response strategy to the strategy profile is a strategy such that

 ~s(u)={0if~{}1+E[∑j≠isj(yj)∣yi=u]E[θ∣yi=u]. (2)

We denote the set of all best responses to a strategy profile by . Finally, we say that a strategy profile is a Bayesian Nash Equilibrium or simply an equilibrium if for all .

An extensively studied model in global games is the case where for all and where are independent and identically distributed Gaussian random variables for some . Furthermore, it is customary to assume that is picked from a non-informative uniform distribution over . See [22] and the references therein for further discussion on this assumption.

In [22], it is shown that there exists a symmetric threshold policy on ’s which corresponds to a Bayesian Nash equilibrium for this instance of global games. In other words, there exists a threshold value such that for , agent chooses to take the risky action and for , she takes the safe action and such an action profile leads to an equilibrium. Here, an important fact is that which means that in such an equilibrium each agent should compare her expected strength of regime given her private observation to a threshold and take a proper action accordingly.

Our Model for Information Structure: In this work, we consider global games with noisy sharing of information. We seek to understand the role of information sharing among the agents in the decision making scenarios that are modeled by global games.

Throughout this work the utility of each agent is given by (1) and the information structure satisfies the following assumption.

###### Assumption 1

We assume that is uniformly distributed over . Also, agent ’s private information is the dimensional random vector:

 yi:=(xi,{yji}j∈[n]∖{i}), (3)

where for all and for , where are i.i.d.  random variables and are i.i.d.  random variables that are independent of , and are given parameters.

The dependence diagram of the random observation vector that is available to agent is shown in Figure 1.

Throughout the rest of the paper, the scalar which is the sum of all the information arrived at agent from the rest of agents plays a central role. We denote this quantity by , i.e.,

 zi:=∑j∈[n]∖{i}yji. (4)

We define a global game with noisy sharing of information as below.

###### Definition 1

A game with utility functions described by (1) and information structure satisfying Assumption 1 is a global game with noisy information sharing.

The major challenge in analyzing global games with noisy information sharing is the limited information of each agent about the underlying fundamental . Note that if all the agents know about the exact realization of , and for either of the symmetric actions or would be appealing. However, in the global games, the perfect knowledge of the underlying fundamental is not available to any agent and each agent has a noisy observation of the fundamental .

## Iii Main Results

In this section, we present the main results of this work including two results. The first result establishes nonexistence of an intuitive equilibrium for global games with noisy sharing of information. The second result shows that under a certain regime, there exists a threshold policy based on each agent’s private information and the (noisy) information that is shared with her by the other agents.

### Iii-a Nonexistence of an Intuitive Equilibrium

In many instances of global games, one can show that there exists a threshold policy based on the expected value of the underlying fundamental given each agent’s information. More precisely, let

 ¯θi(yi)=E[θ∣yi].

It has been shown that if there is no sharing of information (i.e., ) and is uniformly distributed over or has a Gaussian prior, then there exists a threshold value such that the policy is a Bayesian Nash Equilibrium for the global games described above [23].

In the case of global games with noisy sharing of information, one may hope to have a similar result, i.e., there exists a threshold value such that if everyone compares her expected value of the fundamental variable given her own information, the policy would be a Bayesian Nash Equilibrium. For example, in the case of a bank run, one may speculate that if each agent compares her expected strength of the economy given her own information, she should decide whether to take her money out of the bank or not, and this behavior would result in an equilibrium. However, the following result shows that such an intuitive equilibrium for global games with noisy sharing of information does not exist.

###### Theorem 1

. Consider the global game with noisy sharing of information as described in Definition 1. Then, there do not exist threshold values such that the policy is an equilibrium.

### Iii-B Existence of a Threshold Policy Equilibrium

In light of Theorem 1, one may pose the question as how one can extend the existence of threshold policies for conditionally independent signals to the case of interdependent private signals. Here, we show that indeed we can extend such an existence result to the case of global games with sharing of information.

Before presenting this result, the notion of threshold policies is extended from the case of a scalar private information to multi-dimension information vector. Consider the information available to agent , i.e., . For a function , we will be focusing on the symmetric threshold strategies . A strategy is called symmetric if it does not depend on the index of the agents. Namely, in this case, the function is the same for all the agents. We will further limit our attention to the class of functions which are continuous and strictly decreasing with respect to any of the input parameters, while the other input parameters are fixed. Monotone strategies are natural candidates to consider because if taking risky action is not appealing for a given observation, it is natural to consider strategies that assign the safe action to any larger observation. More precisely, let be smaller than component-wise. If agent takes a risky action by observing , it is intuitively expected that it takes the risky action by observing as well. In other words, assuming a threshold strategy , we expect that if , then . A symmetry condition on is also required which will be clarified later in this section.

With this, the main existence results of this paper are stated in the next two theorems.

###### Theorem 2

. Let be a continuous and component-wise strictly decreasing function. Then leads to a threshold policy equilibrium if it is the solution to a certain functional equation characterized by the noise variance parameters of the system.

###### Theorem 3

. For any given , there exists an unbounded set of parameters such that for any there exists a symmetric Bayesian Nash Equilibrium for some continuous and component-wise decreasing function . Furthermore, can be approximated with an arbitrary precision (in norm).

The proof of Theorem 3 will be based on defining a contraction mapping on the space of feasible strategy functions and then utilizing Banach fixed point theorem to establish the existence of that solves the functional equation of Theorem 2. Consequently, it is shown that an iterative method of applying the contraction mapping will result in the convergence to the desired which can be used as an approximation method to find with arbitrary precision.

We will also characterize a sufficient condition on the noise parameters to be contained in the set of Theorem 3. In particular, it is shown that for a fixed ratio of , , for large enough . This implies that the set is unbounded in any direction and the set is bounded in any dierction. Roughly speaking, for a sufficiently noisy environment there exists a threshold type Bayesian Nash Equilibrium.

One may conjecture that Theorem 3 should hold for the whole set of noise parameters and it should be independent from the noise parameters, i.e., . However, we indeed conjecture that such a symmetric equilibrium does not exist for arbitrary .

The above three theorems suggest that in the settings that global games are relevant, one can not be oblivious to the information structure details and only relies on the estimate of the underlying fundamental .

## Iv Proofs

In this section, we provide the proof of the main results discussed in Section III.

The following calculations for the statistics of conditional Gaussian random vectors are needed in the proof of the main results. Next lemma derives the probability distribution of conditioned on the observation vector of the -th agent.

###### Lemma 4

. Suppose that Assumption 1 holds. Let be a scalar such that

 1η2n=1σ2+n−1σ2+τ2.

Further, let and . Then, conditioned on agent ’s observation , is given by

 θ=anxi+bnzi+εθ,

where is a Gaussian random variable independent of .

The proof can be found in Appendix.

To investigate the existence of a threshold policy for agent , we further need to derive the distribution of agent ’s observation vector given agent ’s observation . In a sense, this is agent ’s perception of what is available to agent .

###### Lemma 5

. Let Assumption 1 hold. Further, let

 1γ2n=1τ2+1η2n−1+σ2,

and define and . Then, conditioned on , we can write as jointly Gaussian random variables defined by:

 xk=¯¯¯¯¯xk+εk,  yik=xi+εik,  ylk=¯¯¯¯¯¯ylk+εlk,

where and111Note that and are functions of the observation vector of agent but the dependence is left implicit for brevity.

 {¯¯¯¯¯xk=cnan−1xi+dnyki+cnbn−1∑j∈[n]∖{i,k}yji,¯¯¯¯¯¯ylk=cnan−1xi+dnyli+cnbn−1∑j∈[n]∖{i,l}yji, (5)

and are jointly zero mean Gaussian random variables independent of ; and is independent of .

The proof can be found in Appendix.

### Iv-a Proof of Theorem 1

The underlying idea of the proof can be summarized as follows. Assume to the contrary that is an equilibrium for some , where by Lemma 4. Then the best response strategy of agent , given the strategies of other agents is given by (2). This must be equivalent to by the definition of the Bayesian Nash equlibrium. However, we use the non-singularity of the system of linear equations describing the noise variances to show that this can not happen resulting in a contradiction.

Let be defined as follows:

 βn :=bn+ancnan−1+(n−2)bncnan−1 γn :=andn+(n−2)bncnbn−1 δn :=ancnbn−1+bndn+(n−3)bncnbn−1

Then let be a matrix with the following entries:

 vij=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩an, if i=1,j=1,bn, if i=1,1

In other words, has the following structure:

 V=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝anbnbn…bnβnγnδn…δnβnδnγn…δn⋮⋮⋮⋱⋮βnδnδn…γn⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠.

By Lemma 4 and Lemma 5, we have

 (¯θi(yi),{E[¯θj(yj)∣yi]}j≠i)=VyTi (6)

It is assumed that , i.e., information sharing between agents is noisy and not perfect. This leads to and consequently . We next show that this implies that is non-singular. By subtracting the first column of scaled by from all other columns we get

 V′=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝anbn−δnβnanbn−δnβnan…bn−δnβnanβnγn−δn0…0βn0γn−δn…0⋮⋮⋮⋱⋮βn00…γn−δn⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠.

It can be observed that and hence, by subtracting the first row scaled by from the other rows, is made lower triangular with strictly positive diagonal elements. Therefore, and consequently are non-singular.

Using the fact that is non-singular together with (6) one can find such that , for an arbitrary , and , for , for any arbitrarily large number . Note that the variance of given is a constant function of and (by Lemma 5). Therefore, using Chebyshev’s inequality, for the given threshold values , one can find such that and

 P(¯θj(yj)⩽tj∣yi)⩽ϵ.

But by the structure of an equilibrium (2), we should have

 (n−1)ϵ+1 ⩾∑j≠iP(¯θj(yj)⩽tj∣yi)+1 ⩾¯θi(yi)=ti−ϵ.

Since the above inequality must hold for any , we have .

On the other hand, using the same argument for any , one can find such that and , for , for any arbitrarily large number . Hence,

 ti+ϵ ⩾¯θi(yi) >E(∑j≠i1¯θj(yj)⩽tj∣yi)+1 ⩾1+(1−ϵ)(n−1)=n−(n−1)ϵ.

Since, this holds for any , it follows that which contradicts . Therefore, such a threshold equilibrium does not exist.

The above proof can be extended to a more general case of utility functions that are monotone with respect to the actions of the players and also are continuous functions of the underlying fundamental .

### Iv-B Proof of Theorem 2

We start by introducing the set of threshold policies that we will be focusing on. Such policies are characterized by certain threshold functions. We impose some natural constraints on the threshold function . The description of the best response strategies, formulated in (2), provides a necessary condition on that leads to threshold-type Bayesian Nash equilibrium. This condition can be interpreted as being a fixed point to a certain functional equation resulting from (2). Then we exploit natural properties of , such as being continuous, monotone ,and symmetric, to conclude that such condition is also sufficient for having a threshold-type Bayesian Nash equilibrium characterized by .

Consider the information available to agent , i.e., . For a function , we will be focusing on the threshold strategies . We will further limit our attention to the class of functions which are continuous and strictly decreasing with respect to any of the input parameters, while the other input parameters are fixed. A symmetry condition on is also required which will be clarified later in this section.

The goal is to show there exists, under some conditions on noise parameters , a function with the above conditions, such that the strategy profile is an equilibrium for the global games with noisy sharing of information, where is the threshold policy that prescribes:

 {αi=1  if  h(yi)⩾0αi=0  if  h(yi)<0 (7)

for agent . We refer to such a function as a threshold function, and we refer to the resulting symmetric strategy profile

 s=(1h(y1)⩾0,…,1h(yn)⩾0) (8)

as a symmetric threshold policy. If the strategy profile given in (8) is an equilibrium for the underlying game, we say that the threshold function leads to a (symmetric) threshold policy equilibrium.

By the definition of a (Bayesian Nash) equilibrium, the function leads to a threshold policy equilibrium if a best response of any agent is characterized by the threshold policy described in (7). In other words, we have

 1+∑k∈[n]∖{i}P(h(yk)⩾0∣yi)⩾E(θ∣yi)=anxi+bnzi (9)

if and only if , for any , where is given in (4).

Remark. It is shown in Theorem 1 that an equilibrium with a threshold policy on does not exist. In light of the definition of threshold functions provided here, this result can be stated as the threshold function , where is a constant, does not lead to a threshold policy equilibrium. We emphasize that our definition of a threshold function only considers a symmetric threshold policy (if it exists), where all agents take actions according to the same threshold function .

Consider a continuous threshold function that leads to a Bayesian Nash equilibrium. It can be observed that if , then (9) also turns into equality. In other words, we have

 (10)

Note that given the parameters of the system, the right hand side of (10) can be regarded as an operation on the function . The definition of this operation will be provided later in this section. This motivates us to define a fixed point threshold function as follows:

###### Definition 2

. We say that is a fixed point threshold function if for any choice of , we have if and only if (10) also holds.

We note here that a fixed point threshold function is not necessarily unique. In fact, the set of transition points plays a central role in defining the strategy profile as in (8), rather than itself. Any other strictly decreasing function that has the same set of transition points defines the same strategy profile as does.

For notational convenience, we let

 yi∖k:=(xi,{yji}j∈[n]∖{i,k})

for . Note that if is a continuous fixed point threshold function, then for any and , there exists such that , where is defined in (3). The reason is that the right hand side of (10) is bounded between and , while the left hand side of (10) is unbounded in terms of , while is fixed. Therefore, there exists such that (10) turns into equality and since is a fixed point threshold function, by definition, . We let to denote the solution for . Furthermore, if is strictly decreasing with respect to any of its input parameters, is unique. In fact, is also a strictly decreasing function.

As mentioned before, the function is not unique, however, is unique under some additional conditions. Therefore, instead of explicit characterization of , we will be solving the fixed point equation for and then, we pick as follows:

 h(yi):=Ih(yi∖k)−yki (11)

As discussed earlier, a symmetry condition on with respect to is naturally needed in order to have a symmetric threshold policy equilibrium. In fact all the other agents look the same to the agent and the observations follow the same model, as illustrated in Figure 1, hence the indexing of other agents should not matter. The symmetry condition can be that is the same if is permuted. However, this is too general for our purpose and as constructed in (11) may not satisfy this condition. In fact the symmetry condition only matters when , because the threshold policy will be uniquely determined given the set of solutions for when is strictly decreasing. The symmetry condition is then defined as follows:

###### Definition 3

. We say that is a symmetric threshold function if for any root of , with a permuted is also a root of .

Note that if is strictly decreasing and symmetric fixed point function, the function does not depend on the choice of index .

The following theorem is the main result of this section and presents a sufficient condition on to be a threshold function.

###### Theorem 6

. Let be strictly decreasing and symmetric fixed point threshold function according to Definition 2. Then leads to a threshold policy equilibrium.

###### Proof.

For , let denote the observations of agent . We fix and consider two different cases:

Case 1: .
Note that is a strictly decreasing function and if and only if . Using Lemma 5 we have

 P(h(yk)⩾0∣yi)=P(yik⩽Ih(yk∖i)∣yi),=P(xi+εik⩽Ih(¯¯¯¯¯xk+εk,{¯¯¯¯¯¯ylk+εlk}l∈[n]∖{i,k})), (12)

where and are the means of and conditioned on , respectively, and are derived in (5). Let and denote the joint probability density function (PDF) of the Gaussian random variables . By Lemma 5, is independent of and thus, (12) can be rewritten as in (13), shown on top of the next page.

In (13), is the cumulative distribution function (CDF) of the normal distribution with unit variance. For , let such that , where . Let also and be defined with respect to and as in Lemma 5. Then by following the same arguments and by noting that does not change while changing to ,  (14) follows,

where . Observe that and . Also, note that is a decreasing function and is an increasing function. Therefore, (13) together with (14) imply that

 P(h(yk)⩾0∣yi)⩾P(h(yk)⩾0∣y′i). (15)

By summing (15) over all and using the fixed point property of at as described in Definition 2, we get

 1+∑k∈[n]∖{i}P(h(yk)⩾0∣yi)⩾1+∑k∈[n]∖{i}P(h(yk)⩾0∣y′i)=anxi+bn∑j∈[n]∖{i}y′ji⩾anxi+bnzi=E(θ∣yi). (16)

Therefore, the best response of agent is to take the risky action.
Case 2: .
In this case we take such that . The inequalities in (15) and (16) will be reversed and the best response of agent is to take the safe action. This will complete the proof.

### Iv-C Proof of Theorem 3

In this section, we analyze the convergence of an iterative scheme for finding the threshold function and consequently the threshold policy equilibrium. To this end, a certain operator is defined that captures the update of a threshold policy when agents update their strategy according to the best response strategy rule. Sufficient conditions are derived to guarantee that the operator becomes a contraction mapping. Then the Banach fixed point theorem is utilized to show the convergence of an iterative scheme, that applies iteratively to an initial function, to a fixed-point threshold function. Then Theorem 2 is utilized to show that such threshold function results in a Bayesian Nash equilibrium. This complete the proof of Theorem 3.

As the first step, we find such that the threshold function , as derived in (11), satisfies the conditions of Theorem 6. By replacing with in (10) and using derivation of in (13), (10) can be turned into a fixed point equation for the function . Let

 g(yi∖k):=anxi+bnIh(yi∖k)+bn∑j∈[n]∖{k,i}yji. (17)

Note that is simply the left-hand side of (10). We finally arrive at the definition for the fixed point function as follows.

###### Definition 4

. We call with the input to be a fixed point function if

 g(y)=1+∑l∈[n]∖{i}∫Rn−1φ(Mϵ,lg(y))fϵ(ϵ)dϵ, (18)

where denote the joint probability density function (PDF) of the Gaussian random variables and is defined as

 Mϵ,lg(y)=1bnτ(g(¯¯¯¯¯xl+εl,{¯¯¯¯¯¯yjl+εjl}j∈[n]∖{i,l})−an(¯¯¯¯¯xl+εl)−bn∑j∈[n]∖{i,l}(¯¯¯¯¯¯yjl+εjl)−bnxi), (19)

where and are the means of and derived in (5) in terms of and

 yki=1bng(yi∖k)−anbnxi−∑j∈[n]∖{k,i}yji.
###### Definition 5

. We define the operator to be the operator that maps a sufficiently well-behaved function to defined by

 Tg(y):=1+∑l∈[n]∖{i}∫Rn−1φ(Mϵ,lg(y))fϵ(ϵ)dϵ, (20)

where is defined in (19).

If is a fixed point function, according to Definition 4, then , by definition of in (20). Therefore, finding a fixed point function is equivalent to finding a fixed point for the operator .

In the subsequent discussion, we will derive conditions that will characterize the term sufficiently well-behaved in the above statement. First we notice that can be viewed as a mapping that maps the space of measurable functions to itself. This follows from the fact that for all .

To find a fixed point for the operator , the structure of the fixed point equation (20) suggests the investigation of the iteration

 g(t+1)=Tg(t), (21)

for some sufficiently well-behaved initial function . Indeed, we will prove that induces a contraction mapping on the space and hence, converges to a unique fixed point. Throughout our discussion, is the space of continuous functions from to embedded with the uniform norm:

 ∥g1−g2∥:=supy∈Rn−1|g1(y)−g2(y)|.

Once the fixed point function is found, is derived from according to (17) and then is derived from as in (11). However, we will need to be symmetric according to Definition 3. If a function leads to a symmetric through the mentioned transformations, then is called a quasi-symmetric function. Note that if is quasi-symmetric, then is also quasi-symmetric. Because the definition of suggests that it is independent of the labeling of indices and also there is a symmetry between and , for in (17). Therefore, we limit our attention to the set of quasi-symmetric functions.

Moreover, in order to make sure that leads to a strictly decreasing and consequently a threshold function , we impose a stronger condition on . The Lipschitz continuity is imposed on , where is embedded with the norm. We show that the Lipschitz continuity with parameter is preserved through under certain conditions, i.e., we require that for any :

 |g(yi∖k)−g(y′i∖k)|⩽an|xi−x′i|+an∑j∈[n]∖{k,i}|yji−y′ji|. (22)

Let denote the space of all Lipschitz continuous functions with parameter which are also quasi-symmetric. We aim at characterizing a condition on the noise variance parameters of the system such that the Lipschitz continuity is preserved through the operation . In other words , where .

The following lemma establishes a sufficient condition on the parameters of the system to preserve the Lipschitz continuity through the operation . The Lipschitz continuity of is used to show Lipschitz continuity of , and consequently the Lipschitz continuity of , where the Lipschitz continuity of together with triangle inequality are used. Furthermore, an upper bound on the Lipschitz constant of and consequently on are derived in terms of the Lipschitz constant of and other parameters of the system. That leads to a sufficient condition for not increasing the Lipschitz constant through the operation , described in the following lemma. Let

 en:=max{cnan−1,dn,cnbn−