Social Welfare Maximization and Conformism via Information Design in Linear-Quadratic-Gaussian Games
SSocial Welfare Maximization and Conformism via InformationDesign in Linear-Quadratic-Gaussian Games
Furkan Sezer, Hossein Khazaei, and Ceyhun Eksin ∗ February 26, 2021
Abstract
We consider linear-quadratic Gaussian (LQG) games in which players have quadratic payoffs thatdepend on the players’ actions and an unknown payoff-relevant state, and signals on the state that followa Gaussian distribution conditional on the state realization. An information designer decides the fidelityof information revealed to the players in order to maximize the social welfare of the players or reducethe disagreement among players’ actions. Leveraging the semi-definiteness of the information designproblem, we derive analytical solutions for these objectives under specific LQG games. We show that fullinformation disclosure maximizes social welfare when there is a common payoff-relevant state, when thereis strategic substitutability in the actions of players, or when the signals are public. Numerical resultsshow that as strategic substitution increases, the value of the information disclosure increases. Whenthe objective is to induce conformity among players’ actions, hiding information is optimal. Lastly, weconsider the information design objective that is a weighted combination of social welfare and cohesivenessof players’ actions. We obtain an interval for the weights where full information disclosure is optimalunder public signals for games with strategic substitutability. Numerical solutions show that the actualinterval where full information disclosure is optimal gets close to the analytical interval obtained assubstitution increases.
In an incomplete information game, multiple players compete to maximize their individual payoffs thatdepend on the action of the player, other players’ actions and on the realization of an unknown state.Incomplete information games are used to model power allocation of users in wireless networks with unknownchannel gains [1, 2, 3, 4], traffic flow in communication or transportation networks [5, 6, 7], oligopoly pricecompetition [8, 9], consumer behavior in a demand response management scheme [10, 11], coordination ofautonomous teams [12, 13], and currency attacks of investors [14, 15]. The information design problemrefers to the determination of the information fidelity of the signals given to the players so that the inducedactions of players maximize a system level objective. In this problem, we can envision the existence of aninformation designer that can provide the “best” information about the payoff-relevant unknown states tothe players according to its objective. As per the above examples, the information designer can representan entity such as a system designer overseeing the spectrum allocation, a market-maker, an independent ∗ Authors are with the Department of Industrial and Systems Engineering, Texas A&M University, College Station, TX77843 USA (e-mails: [email protected], [email protected], [email protected]). This work was supported by NSF CCF-2008855. a r X i v : . [ m a t h . O C ] F e b ystem operator in the power grid, system designer, or the federal reserve. Depending on the context, theinformation designer may have different objectives such as maximizing social welfare, inducing consensusbetween players’ actions, or reducing the likelihood that players take a certain (undesired) action. In theabsence of a literal information designer, the information design problem can be used to analyze the extent towhich extra information to the players can influence the actions of players and affect a system level objective[16, 17].We focus on the information design problem in linear quadratic Gaussian (LQG) games. In an LQGgame, the players have quadratic payoff functions, and the state and the signals (types) come from a Gaussiandistribution [18]. The quadratic payoff structure implies that there exists rational linear strategies whichare mappings from local signals to actions. Under certain assumptions, the rational behavior in LQG gamesdefined as the Bayesian Nash equilibrium (BNE) is unique. The linearity of BNE strategies allow theinformation design problem to be a semi-definite program (SDP) when the information designer’s objectiveis a quadratic function of the action profile of the players and the payoff-relevant states [19] (Section 2).Building on the SDP formulation of the information design problem established in [19], we analyze optimalinformation structures when the system level objective is to maximize social welfare while maintaining acertain level of conformism in the actions of the players. Social welfare is defined as the sum of players’payoffs. We measure the conformism of the players by the total deviation of the actions from the meanaction. Minimizing the total deviation between the actions of players induces the actions of players to beclose to each other. We represent the objective to maximize social welfare while maintaining conformism ofthe players by a weighted combination of the social welfare and the total deviation of actions objectives.An information structure comprises signal transmission rules and the probability distribution from whichsignals are generated. Signals transmitted to players convey information about payoff relevant states. Privacylevels of information structures form a distinction between them. The information structure is public when allplayers receive a common signal. Otherwise, when the players receive individual signals, the signal structureis private . Another distinction is based on the fidelity of information carried by the signals. A signal cancarry no , partial or full information . No information disclosure does not improve the prior information of theplayers about the payoff relevant state, while signals reveal the payoff relevant state under full informationdisclosure. A partial information disclosure is when the signals carry some information but do not fullyreveal the payoff relevant state to the players.When the information designer’s objective is to maximize social welfare, we show that full informationdisclosure is optimal if there is a common payoff state (Proposition 7), when there are symmetric interactionsbetween the actions of players, or if we restrict feasible set to the set of public information structures(Proposition 8). In particular, when the interactions are strategically substitutable, i.e., a player’s incentiveto increase its action decreases when others increase their actions, full information disclosure is optimalfor welfare maximization (Theorem 1). Conversely, when the interactions are strategic complementary,i.e., a player’s incentive to increase its action increases when others increase their actions, full informationdisclosure is optimal for welfare maximization only if the complementarity is weak. These results follow theintuition that the designer would like to reveal as much information as possible when positively correlatingthe actions of players improves the system level objective, i.e., when the payoffs of players are aligned withthe system-level objective [17].When the objective is to minimize the deviation of players’ actions, we show that no information disclosureis optimal for any LQG game (Proposition 9). That is, players’ actions are closer to consensus wheninformation is hidden from the players.If the information designer aims to maximize social welfare while maintaining a level of conformismwithin the society, we identify a critical weight on the conformism term of the objective based on payoffcoefficients below which full information disclosure is preferred to no information disclosure (Propositions 102nd 11). That is, the benefit of revealing information outweighs the cost of deviation between players’ actions.Numerical solutions to the information design problem show the existence of private signal distributions thatoutperform both public (full and no) information disclosure schemes. Information design is commonly used to induce desirable behavior in congestion games with a particularfocus on routing or flow problems in transportation networks [20, 21, 22, 7]. Indeed, a well-establishedcommon result in these works is the diminishing value of extra information, and individual players andthe system level being worse-off under full information compared to private information schemes. Besidescongestion control, information design framework is used to identify key/central players in social networkswith respect to the goals of the system designer [23, 24], to maximize the utility of the insured agents in acompetitive insurance market [25], and to design public health warning policies against recurrent risks, e.g.,pandemics, for public health agencies with different reputation levels [26]. Here, instead of focusing on aparticular game, we study the class of LQG games that have broad applications in various fields includingoperations, autonomous teams, and smart grids.In general, information design problem involves a maximization over all information structures (all pos-sible distributions) which is algorithmically challenging to solve. In an effort to obtain analytical solutions,and thus insights about the effects of the information design on the system, current approaches make struc-tural assumptions about the state space (finite, Gaussian), the system designer’s objective (depends onposterior belief induced or on the actions induced, or is a step function of the induced posterior mean), andthe payoff of agents (posterior mean induced by a public signal)—see [27] for a more detailed discussionof different structural assumptions on information design problems. Similarly, we consider linear-quadraticutility functions for the agents, a quadratic objective for the system designer, and restrict the informationstructure to Gaussian signals as in [19]. This structure allows the representation of the information designproblem as a tractable optimization problem, in particular an SDP with a quadratic objective and linearconstraints. As per the discussion above, by focusing on particular objectives for the designer (social welfareand conformism), we provide analytical insights about the value of information and optimal informationstructures.Information design is one form of intervention mechanism to influence behavior of strategic agents [6,28]. Other intervention mechanisms include providing financial incentives [6, 29, 30], system utility design[31, 32, 33, 34, 35], and nudging or player control during learning dynamics [36, 37, 38, 39, 40]. Financialincentives include taxations [6] or rewards [29] that directly modify the utility of the agents. System utilitydesign is motivated by the use of game theoretic learning dynamics in technological multi-agent systemsin which individual payoffs are assigned/coded by a system designer prior to the dynamics so that thelearning dynamics converge to a desired operating conditions. Relevant to concepts in seeding in advertisingand recommendation systems, nudging or player control primes a few individuals to follow the systemdesigner’s will or intervene with the outcomes of a few individuals, so that the learning dynamics convergeto desired equilibria. In contrast to these line of work, the information design framework aims to managethe uncertainties of players so that their expectation of their payoffs align with the objective of a systemdesigner. In information design, the system designer does not control agents directly, rather it determineswhen and who to reveal information to, so that players’ evaluation of their payoffs lead to better outcomesfrom the system designer’s perspective. In this sense, there is a limit to the system designer’s capability toachieve its goal. This limit determines the value of information.3
Information Design in Linear-Quadratic-Gaussian (LQG) Games
A linear-quadratic-Gaussian (LQG) game corresponds to an incomplete information game with quadraticpayoff functions and Gaussian information structures. For a given set of players denoted with N = { , ..., n } ,each player i ∈ N decides on his action a i ∈ A i ≡ R according to the payoff function, u i ( a, γ ) = − H i,i a i − (cid:88) j (cid:54) = i H i,j a i a j + 2 γ i a i + d i ( a − i , γ ) (1)where a ≡ ( a i ) i ∈ N ∈ A ≡ R n and γ ≡ ( γ i ) i ∈ N ∈ Γ ≡ R n correspond to an action profile and a payoff state,respectively. The term d i ( a − i , γ ) is an arbitrary function of the opponents’ actions a − i ≡ ( a j ) j (cid:54) = i and payoffstate γ . We collect the coefficients of the quadratic payoff function in a matrix H = [ H i,j ] n × n .Payoff state γ follows a Gaussian distribution, i.e., γ ∼ ψ ( µ, Σ) where ψ is a multivariate normal prob-ability distribution with mean µ ∈ R n and covariance matrix Σ. Payoff structure of the game is defined as G ≡ (( u i ) i ∈ N , ψ ). Each player i ∈ N receives a private signal ω i ∈ Ω i ≡ R m i for some m i ∈ N + . We definethe information structure of the game ζ ( ω | γ ) as the conditional distribution of ω ≡ ( ω i ) i ∈ N given γ . Weassume the joint distribution over the random variables ( ω, γ ) is Gaussian; thus, ζ is a Gaussian distribution.Next, we provide two canonical examples of LQG games. Firms determine the prices for their goods ( p i ) facing a marginal cost of production ( γ i ). The demandfunction is a function of the price of all the firms, q i = a − bp i + (cid:80) j (cid:54) = i p j with a and b positive constants.The payoff function of the firm i is its profit given by its revenue q i p i minus the cost of production γ i q i , u i ( a, γ ) = q i p i − γ i q i . (2)The Bertrand competition model is analyzed in [8]. When firms choose their production quantity, insteadof determining their price, and the price is determined by a linear inverse demand function, the LQG gameis called a Cournot competition. Payoff functions of players under common payoff state γ representing, e.g., a stock value, are given by u i ( a, γ ) = − (1 − β )( a i − γ ) − β ( a i − ¯ a − i ) , (3)where β ∈ [0 ,
1] and ¯ a − i = (cid:80) j (cid:54) = i a j / ( n −
1) represents the average action of other players. The first termin (3) denote the players’ urge for taking actions close to the payoff state γ . The second term accounts forplayers’ tendency towards taking actions in compliance with the rest of the population. The constant β gauges the importance between the two terms. The payoff captures settings where the valuation of a good,e.g., stock, depends not just on the performance of the company but also on what other players think aboutits value [15]. 4 .2 Bayesian Nash and Correlated Equilibria A strategy of player i maps each possible value of the private signal ω i ∈ Ω i to an action s i ( ω i ) ∈ A i ,i.e., s i : Ω i → A i . A strategy profile s = ( s i ) i ∈ N is a Bayesian Nash equilibrium (BNE) with informationstructure ζ , if it satisfies the following inequality E ζ [ u i ( s i ( ω i ) , s − i , γ ) | ω i ] ≥ E ζ [ u i ( a (cid:48) i , s − i , γ ) | ω i ] , (4)for all a (cid:48) i ∈ A i , ω i ∈ Ω i , i ∈ N where s − i = ( s j ( ω j )) j (cid:54) = i is the equilibrium strategy of all the other players,and E ζ is the expectation operator with respect to the distribution ζ and the prior on the payoff state ψ .The following result states a sufficient condition for having an unique BNE strategy, and provides a set oflinear equations to determine the coefficients of the linear BNE strategy. Proposition 1 ([18]) . Suppose that H + H T and var( ω i ) are positive definite for each i ∈ N . Then LQGgame has a unique Bayesian Nash equilibrium given by s i ( ω i ) = ¯ a i + b Ti ( ω i − E ζ [ ω i ]) for i ∈ N, (5) where b , ....b n are determined by the following systems of linear equations: (cid:88) j ∈ N H i,j cov ( ω i , ω j ) b j = cov ( ω i , γ i ) for i ∈ N, (6) and cov ( · , · ) represents the covariance between two random variables. Assumptions of Proposition 1 guarantee existence and uniqueness of a linear strategy (5) that satisfy (4)with coefficients (6). We assume the sufficient conditions above throughout the paper.An action distribution represents the probability of observing an action profile a ∈ A when agents followa strategy profile s under ζ . We define the action distribution as φ ( a | γ ) = (cid:80) ω : s ( ω )= a ζ ( ω | γ ). A Bayesiancorrelated equilibrium (BCE) is an action distribution in which no individual would profit by unilaterallydeviating from selecting actions according to the given action distribution. The formal definition follows. Definition 1.
An action distribution φ under ζ is a BCE if and only if it satisfies E φ [ u i (( a i , a − i ) , γ ) | a i ] ≥ E φ [ u i (( a (cid:48) i , a − i ) , γ ) | a i ] (7) for all a i , a (cid:48) i ∈ A i and i ∈ N where E φ [ ·| a i ] is the conditional expectation with respect to the action distribu-tion φ and information structure ζ given action a i ∈ A i . An equilibrium action distribution φ , corresponding to a BNE strategy profile s under ζ , i.e., φ ( a | γ ) = (cid:80) ω : s ( ω )= a ζ ( ω | γ ), satisfies (7) as stated in the following result. Proposition 2 (Corollary 2, [41]) . An equilibrium action distribution is a BCE under any informationstructure. If a BCE corresponds to an equilibrium action distribution, a corresponding information structureexists.
Using Propositions 1 and 2, we can derive a necessary and sufficient condition for an action distributioncomprised of jointly normally distributed action profile and payoff state.5 roposition 3 ([19]) . An action distribution φ comprised of jointly normally distributed action profile anda payoff state is a BCE if and only if the following conditions hold E φ [ a ] = a (8) (cid:88) j ∈ N H i,j cov ( a i , a j ) = cov ( a i , γ i ) . (9)Solution of (5) and (6) by b i = 1 and a i = E ζ [ ω i ] constitute a necessary and sufficient condition for aBCE by Proposition 1. Conditions given in (8) and (9) corresponds to this solution; thus, Proposition 3 isestablished. An information designer aims to optimize the expected value of an objective function f ( a, γ ) that is quadraticin its arguments by deciding on an information structure ζ from a feasible region Z. Information designerwill follow the timeline given below:1. Selection of ζ ∈ Z and notification of all players about ζ.
2. Realization of γ and subsequent draw of signals w i , ∀ i ∈ N from ζ ( ω, γ ) .
3. Players act according to BNE under ζ. The optimization problem of the information designer is as follows:max ζ ∈ Z E ζ [ f ( s, γ )] (10)where s is the unique BNE under ζ . Using the equilibrium action distribution φ under ζ and Proposition2, we can replace the objective in (10) with E φ [ f ( a, γ ]. Thus, the information design problem in (10) isequivalent to the following optimization problem [19]:max φ ∈ C ( Z ) E φ [ f ( a, γ )] (11)where C ( Z ) = { φ : φ is the equilibrium action distribution under ζ ∈ Z } . (12)The solution to (11) and (12) determines a distribution ζ for signals conditional on the state γ . Uponrealization of γ , information designer draws signals from ζ and shares them with the players. Note that wedo not assume that the information designer knows the state realization. It just needs to be able to generatesignals conditioned on the realization of the state.When the objective function is quadratic, we can rewrite (11) as a Frobenius inner product of thecoefficient matrix F and the covariance matrix X of ( a, γ ) under φ ∈ C ( Z )—see [19] for details of thederivation. That is, max X ∈X ( Z ) F • X := max X ∈X ( Z ) (cid:20) F , F , F , F , (cid:21) • (cid:20) var ( a ) cov ( a, γ ) cov ( γ, a ) var ( γ ) (cid:21) (13)6here X ( Z ) = { X ∈ P n + } and F • X := n (cid:88) i =1 2 n (cid:88) j =1 F i,j X i,j , (14)and P n + represents the set of all 2 n × n symmetric positive semi-definite matrices. In the objective (13),the variance matrix of actions var ( a ), and covariance matrix of the actions and the payoff state cov ( γ, a )make up the decision variable X . We can assume F , is an n by n zero matrix O because the variance ofthe payoff state var ( γ ) does not depend on the information structure ζ .Note in (13), we do not restrict the covariance matrix X to belong to the covariance matrices induced byrational actions of individuals, i.e., (12) is not enforced. We introduce the following set of constraints thatassure the covariance matrix X is induced by an equilibrium action distribution: (cid:88) j ∈ N H i,j cov ( a i , a j ) = cov ( a i , γ i ) . (15)Lastly, we assign the payoff state covariance matrix var ( γ ) to the corresponding elements of X , var ( γ ) = [ cov ( γ i , γ j )] nxn = [ x n + i,n + j ] nxn . (16)Together constraints (14)-(16) ensure that X is a feasible covariance matrix of an equilibrium action distri-bution. We state the information designer’s optimization problem using matrix notation:max X ∈X ( Z ∗ ) F • X (17)s.t. M k,l • X = cov ( γ k , γ l ) , ∀ k, l ∈ N with k ≤ l (18) R k • X = 0 ∀ k ∈ {
1, . . , n } (19) X ∈ P n + (20)where M k,l = [[ M k,l ] i,j ] nx n ∈ P n and R k = [[ R k ] i,j ] nx n ∈ P n are defined as[ M k,l ] i,j = / k < l, i = n + k, j = n + l / k < l, i = n + l, j = n + k k = l, i = n + k, j = n + l R k ] i,j = H k,k if i = j = k,H k,j / if i = k, ≤ j ≤ n, j (cid:54) = k, − / if i = k, j = n + k,H k,i / if j = k, ≤ i ≤ n, i (cid:54) = k − / if j = k, i = n + k, Z yielding an optimal information structure ζ ∗ . Action distribution representation isused to restate the original problem as the maximization of the designer’s objective by selecting an actiondistribution φ ∗ ( a, γ ) as per Propositions 2 and 3. This problem in (11) is then equivalent to selecting acovariance matrix of the action profile and the payoff states X (13) when the objective function has thequadratic form. Thus the original information design problem (10) is transformed to the maximization of alinear function of a positive semi-definite matrix X subject to linear constraints. The problem (17) - (20),formulated in [19], is a SDP.Next we define an important special case of the above problem where we restrict the information structureto public signals. That is, all players receive the same signal, and it is common knowledge that they willreceive the same signal. A formal definition follows. 7 efinition 2 (Public Information Structure) . An information structure which has ω = .... = ω n withprobability one is called a public information structure. Z p ⊆ Z denotes the set of public informationstructures. We define two important feasible solutions to (17) - (20) (no information and full information disclosure).
Definition 3 (No information disclosure) . No information disclosure refers to the case when there is noinformative signal sent to the players. In this case, the equilibrium action of players is given by H − µ . Theinduced decision variable and the objective value is given by X = (cid:20) O OO var ( γ ) (cid:21) and F • X = 0 . (21) Definition 4 (Full information disclosure) . The signals sent to the players reveal all elements of payoff state γ under full information disclosure. Equilibrium actions is given by H − γ . The induced decision variable X and the objective value F • X is given by X = (cid:20) H − var ( γ )( H − ) T H − var ( γ ) var ( γ )( H − ) T var ( γ ) (cid:21) (22) and F • X = F H • var ( γ ) (23) where F H = ( H − ) T ( F + F H + H T F ) H − . (24)Now we state conditions for the optimality of no and full information disclosure solutions, when werestrict the information design problem to public information structures. Let var ( γ ) = DD T such that D isan n × k matrix of rank k where k is the rank of var ( γ ) . Proposition 4 ([19]) . Assume D T F H D (cid:54) = O is negative semi-definite. Then, no information disclosure isoptimal in Z p , and full information disclosure is not optimal in Z ∗ . Proposition 5 ([19]) . Assume D T F H D (cid:54) = O is positive semi-definite. Then, full information disclosure isoptimal in Z p , and no information disclosure is not optimal in Z ∗ . Social welfare is the sum of individual utility functions, which is stated as follows: f ( a, γ ) = n (cid:88) i =1 u i ( a, γ ) = n (cid:88) i =1 ( − H i,i a i − (cid:88) j (cid:54) = i H i,j a i a j + 2 γ i a i + d i ( a − i , γ )) . (25)Given the objective, the coefficient matrix F is as follows, F i,j = − H i,i i ≤ n, j ≤ n, i = j − H i,j i ≤ n, j ≤ n, i (cid:54) = j n ≤ i ≤ n, j ≤ n, i − n = j n ≤ j ≤ n, i ≤ n, i = j − n H and identity matrix as sub-matrices, F = (cid:20) − H II O (cid:21) . (27)Our first result shows that full information disclosure will always be preferred to no information disclosurein social welfare maximization. Proposition 6. If H (cid:31) , then full information disclosure never performs worse than no informationdisclosure for maximizing social welfare objective.Proof. No information disclosure has the objective value F • X = 0 regardless of F. Objective value of fullinformation disclosure is defined in (23) as F • X = F H • var ( γ ). Covariance matrices are in general positivesemi-definite; therefore, var ( γ ) (cid:23) . If F H (cid:31)
0, then F • X = F H • var ( γ ) ≥ . We need F H = H − (cid:31)
0. If H is positive definite, then H − is positive definite. ( Theorem 7.6.1, [42]). Thus, H − (cid:31) H (cid:31) H automatically holds when the payoff issymmetric given the sufficient condition for the existence and uniqueness of a BNE in Proposition 1. Theresult implies that no information disclosure cannot be an optimal information structure for social welfaremaximization.Next, we show that full information disclosure is the optimal solution to the social welfare maximizationproblem for some important special cases. We consider a scenario in which the payoff states are identical, i.e., γ = γ = . . . = γ n . In this setting wehave the following result. Proposition 7.
Assume H is symmetric and γ i = γ j , ∀ i, j ∈ N . Then, full information disclosure isoptimal for social welfare maximization objective.Proof. First we note that the objective function f in (27) is such that F i,n + j = 0 for ∀ i, j ∈ N with i (cid:54) = j .Moreover, we have F n + i,n + j = 0 , ∀ i, j ∈ N . Therefore, F • X = n (cid:88) i =1 n (cid:88) j =1 F i,j cov ( a i , a j ) + 2 n (cid:88) i =1 F i,n + i cov ( a i , γ i ) . (28)Using the BCE condition in (9) for the corresponding terms in (28), we obtain F • X = n (cid:88) i =1 n (cid:88) j =1 ( F i,j + 2 F i,n + i H i,j ) cov ( a i , a j ) . (29)Thus we have F • X = E • var ( a ) where the coefficients of the matrix E is given as E i,j = F i,j + F i,i + n H i,j + F j,n + j H j,i ∀ i, j ∈ N. (30)Substituting in the coefficients of the objective function in (27), we get that E = H T . Since H is symmetric, E = H . From Proposition 9 in [19], we have that if E = κH for some constant κ >
0, then full informationdisclosure is optimal under common payoff states. In our setting, the condition holds with κ = 1.9roposition 7 establishes that full information disclosure is the optimal information structure if the payoffstate is common and H is symmetric. In the following example, we analyze the optimal information structurefor social welfare maximization when the payoff states are correlated, but not necessarily common, and thegame payoff matrix H is asymmetric. Example (Asymmetric payoffs and correlated payoff states):
We define an asymmetric payoff matrix H by determining its off-diagonal elements using a uniformly distributed random variable U i,j , for i, j ∈ N with range [ − , H i,j = (cid:40) , i = j, where i, j = 1 , , ..n cU i,j , i (cid:54) = j, where i, j = 1 , , ..n (31)where c ∈ [0 ,
1] is a constant determining the magnitude of the asymmetry.Figure 1 shows the suboptimality of full information disclosure as we vary the correlation betweenpayoff states
Corr ( γ i , γ j ) between 0.5 and 1 under several values for the magnitude of asymmetry c ∈{ . , . , . , . , } . Note that when Corr ( γ i , γ j ) = 1, there is a common payoff state. The loss with respectto the optimal information structure under full information disclosure increases with growing asymmetryand decreasing correlation. The rate of increase in the percentage difference with respect to decreasingcorrelation is larger when asymmetry is larger in magnitude, e.g., compare lines associated with c = 1 and c = 0 . Figure 1:
Percentage difference between optimal objective value and objective value of full information disclosureversus correlation between payoff states. We consider an asymmetric submodular game with payoffs given in (31).The loss with respect to the optimal information structure under full information disclosure increases with growingasymmetry and decreasing correlation. .2 Submodular and supermodular games Next, we derive the optimal solution for a subset of submodular and supermodular games. A game issubmodular if ∂ U i ∂a i a j < , ∀ i (cid:54) = j ∈ N . Otherwise, ∂ U i ∂a i a j > , ∀ i (cid:54) = j ∈ N , the game is supermodular. In aquadratic game, these partial derivatives depend on the off-diagonal elements of the payoff matrix H , i.e, ∂ U i ( a, γ ) ∂a i a j = − H i,j . (32)In a submodular game, a player’s marginal utility of “increasing” its action decreases with increases inother players’ actions. This strategic interaction between players’ actions and payoffs is known as strategicsubstitutability. In contrast, the payoff from increasing its action value increases as the actions of otherplayers increase in a supermodular game. This strategic interaction between players’ actions and payoffs isknown as strategic complementarity. Bertrand competition and beauty contest games are both examplesof supermodular games. In Bertrand competition, if player j increases its price, the demand for player i ’sproduct increases yielding a higher payoff for player i . In the beauty contest game, an increase in the actionof a player, increases the incentive for others to evaluate their evaluations.We state our main result for submodular and supermodular games assuming the payoff matrix H hasthe following form H i,j = (cid:40) i = j ; i, j ∈ { , , ..n } h if i (cid:54) = j ; i, j ∈ { , , ..n } . (33) Theorem 1.
Assume (cid:80) ni =1 var ( γ i ) ≥ h (cid:80) i (cid:54) = j cov ( γ i , γ j ) and H has the form in (33) . Then, full informa-tion disclosure is optimal for the following games given the social welfare maximization objective:1. A submodular game such that < h <
2. A supermodular game such that − n − < h < .Proof. See Appendix for the proof.Theorem 1 shows that full information disclosure is optimal for welfare maximization when the game issubmodular. That is, social welfare maximization objective is aligned with the incentives of players whenincreasing one player’s action reduces the incentive for other players to increase their actions. In contrast,when increasing one player’s action increases the incentive for other players to increase their actions, i.e.,when we have a supermodular game, the optimality of full information disclosure is optimal as long as theeffect of another players’ actions on a player’s action h is small. Indeed, full information disclosure ceases tobe optimal in supermodular games as the number of players increases.Another sufficient condition for optimality of full information disclosure in Theorem 1 is the diagonaldominance of the covariance matrix of the payoff state. In the following numerical example, we identify thatthe full information disclosure remains optimal even when the diagonal dominance assumption does not holdin symmetric submodular/supermodular games. Example (Relaxing the diagonal dominance of var ( γ ) ): We consider a symmetric submodular gameamong n = 4 players with payoff coefficent matrix H given in (33). We assume var ( γ ) has the followingform. var ( γ ) ( i,j ) = (cid:40) v, if i = j ; i, j = 1 , , , . , if i (cid:54) = j ; i, j = 1 , , , . (34)11 .4 0.42 0.44 0.46 0.48012345 0.4 0.42 0.44 0.46 0.4800.511.522.533.5 (a) submodular ( h >
0) (b) supermodular ( h <
Comparison of the social welfare values under full information and no information disclosure. The gamesconsidered in (a) and (b) are symmetric submodular and supermodular games with payoffs given in (33). Thecovariance matrix of the payoff states is given by (34). As the uncertainty in the payoff states increase, the benefitof (value of) full information disclosure compared to no information disclosure increases.
When we compare the social welfare value under full information disclosure solution (22) with the optimalsolution to the information design problem in (17)-(20), we find that they are identical for all values of v ∈ [0 . , . v , the diagonal dominance assumption is not satisfied. We considered theoff-diagonal elements of the payoff matrix values h ∈ {− . , − . , − . , . , . , . , . } for Fig. 2. Thisexample suggests that full information disclosure remains optimal even when the diagonal assumption is notsatisfied for both submodular and supermodular games. Fig. 2 shows the increasing gap between the socialwelfare values under full and no information disclosure. Indeed, as the dependence of the payoffs on otherplayers’ actions, i.e., | h | , increases, objective value increases. This means the value of revealing informationbecomes more important as strategic complementarity or substitutability increase. Restricting the information structure to public signals, we establish that full information disclosure is optimalfor social welfare maximization.
Proposition 8.
Assume H is positive definite and consider Z p as the feasible set is the set of publicinformation structures. Then, full information disclosure maximizes social welfare.Proof. H is positive definite, thus H − is positive definite by ([42], Theorem 7.6.1). Therefore, K T F H K (cid:54) = 0is positive definite for any matrix K where F H = ( H − ) T ( − H + IH + H T I ) H − = H − . Thus, the result follows from Proposition 5. 12ogether with the previous results in this section, Proposition 8 implies that in scenarios where fullinformation disclosure is not optimal, e.g., for supermodular games, the optimal information structure hasto include private signals.
We consider a scenario in which the information designer would like to make the actions of the playersas close to each other as possible, i.e., conform to widely accepted behavior. We represent the designer’sobjective using the sum of squared deviation between players’ actions and mean action ¯ a , f ( a, γ ) = − n (cid:88) i =1 ( a i − ¯ a ) , where ¯ a = 1 n n (cid:88) i =1 a i . (35)This objective is relevant in scenarios where deviation of the players’ actions can cost resources, e.g., in anenergy distribution setting [43].We can represent the objective function above as the Frobenius product of a coefficients matrix F andthe covariance matrix X , i.e., F • X , where F i,j = (1 − n ) /n if i = j, i ≤ n /n if i (cid:54) = j, i ≤ n, j ≤ n Proposition 9.
No information disclosure is a maximizer of the objective function in (35) .Proof.
When we check eigenvalues of F, we see it has n − n + 1 eigenvalueswith value of 0 . Thus, F is negative semi-definite. We know X is positive semi-definite. We deduce that F • X ≤ . Objective value of no information disclosure is 0 by (21); thus, no information disclosure isoptimal.Proposition 9 implies that the information designer induces the maximum similarity between players’actions by not revealing any information to the players. Broadly, hiding information from players is optimalwhen there is a conflict between the utility functions of the players and the information designer’s objective.
We consider the following optimization problemmax X ∈X ( Z ) ((1 − λ ) F sw + λF ssd ) • X, λ ∈ [0 ,
1] (37)where F sw and F ssd refers to objective coefficient matrices for social welfare objective (26) and sum ofsquared deviations (36), respectively. F := (1 − λ ) F sw + λF ssd is in the form below F = (cid:20) Y (1 − λ ) I (1 − λ ) I O (cid:21) , (38)13here the elements of Y are defined as Y i,j = (cid:40) λ (1 − n ) n − (1 − λ ) H i,j if i = j, for i ∈ N, j ∈ N λn − (1 − λ ) H i,j if i (cid:54) = j, for i ∈ N, j ∈ N. (39)In (37), the designer has a trade-off between maximizing welfare and inducing conformism among theactions of the players in the system. The constant λ quantifies the importance of coherence between players’actions. Our theoretical results for this multi-objective maximization problem in (37) focus on submodulargames in which the payoff coefficients matrix has the form in (33) with coefficient h ∈ (0 , Common payoff state:
We consider identical payoff states, i.e., γ = γ = · · · = γ n . Proposition 10.
Assume H has the form of (33) with h ∈ (0 , and γ i = γ j , ∀ i, j ∈ N , and λ < − h − h and λ ∈ (0 , . (40) Then, no information disclosure is not optimal for the problem defined in (37).Proof.
We first write down E as defined in (30) for this objective. E i,j = (cid:40) λ (1 − n ) n + 1 − λ if i = j λn + (1 − λ ) h if i (cid:54) = j (41)First eigenvalue of E is equal to [( n − h + 1](1 − λ ). The rest of the eigenvalues of E are equal to − λ + (1 − λ )(1 − h ). E is positive definite if both eigenvalues are greater than zero. The assumptions in (40)ensure that E is positive definite. If E is positive definite, then the objective value E • X = E • var ( a ) (cid:23) λ isbelow some threshold. As h → + , we have the region of no optimality increase to λ ∈ (0 , . h ) increases. Indeed wehave the region of no optimality given by λ ∈ (0 , (cid:15) (cid:15) ) when h = 1 − (cid:15) for small (cid:15) >
0. That is the dominance ofwelfare maximization in dictating the information designer disappears as strategic substitutability increases.
Public information structures:
We restrict our attention to the set of public information structures Z p ⊆ Z in which each player receives the same signal. Proposition 11.
Assume H has the form in (33) with h ∈ (0 , , feasible set is public information structures Z P , and λ < − h − h and λ ∈ (0 , . (42) Then, full information disclosure is optimal for the problem given in (37) .Proof.
We start by writing out F H as defined in (24) F H = ( H − ) T ( F + F H + H T F ) H − (43)= ( H − ) T ( Y + (1 − λ ) IH + H T (1 − λ ) I ) H − (44)= ( H − ) T ( Y + 2(1 − λ ) H ) H − . (45)14 H (cid:31) Y + 2(1 − λ ) H (cid:31) T = Y + 2(1 − λ ) H . Thus, T i,j = (cid:40) λ (1 − n ) n + (1 − λ ) if i = j λn + (1 − λ ) h if i (cid:54) = j (46)for all i, j ∈ N. We need the conditions below for T (cid:31) . T is the same as E defined in (41). Therefore, thefirst eigenvalue of T is [( n − h +1](1 − λ ) and the rest of the eigenvalues of T are equal to − λ +(1 − λ )(1 − h ).The matrix T is positive definite if both of the eigenvalues are greater than zero. The conditions in (42)ensure that T is positive definite. Thus by Proposition 5, full information disclosure is optimal for publicinformation structures.Proposition 11 determines the threshold of the constant λ below which social welfare maximization isdominant under public information structures. As shown in Proposition 8, full information disclosure isoptimal if the objective is social welfare maximization and feasible set is the set of public informationstructures. Similarly, full information disclosure is optimal for the multi-objective case if social welfaremaximization objective is important enough.It is worth noting that the conditions we seek in Proposition 10 and Proposition 11 are the same. Thisstems from the fact that we seek the same matrix (see E in (41) and T in (46)) to be positive definite. Alongthe lines of discussion following Proposition 10, we have the region of optimality for full information shrinkas strategic substitutability increases.We note that the threshold for λ below which full information disclosure is optimal in (42) is a sufficientcondition. We assess the strictness of this threshold and the optimality of no and full information disclosuresfor the class of general information structures in a numerical example. Numerical example:
We consider a symmetric game with the payoff matrix H having the form in (33)among n = 4 players. We consider h = 0 . , . var ( γ ) be as follows var ( γ ) ( i,j ) = (cid:40) , if i = j ; i, j = 1 , , , , if i (cid:54) = j ; i, j = 1 , , , . (47)According to Proposition 11, λ region for the optimality of full information disclosure is given by { (0 , ) , (0 , ) , (0 , ) } for h ∈ { . , . , . } , respectively. These λ thresholds are marked by purple dashedline in Figure 3. Black star shows the intersection point of full information disclosure and no informationdisclosure. Figure 3 shows that the region of λ for the optimality of full information under public informa-tion structures is larger in the example than the theoretical regions. The gap between the optimality regionguaranteed by Proposition 11 and real region of optimality decreases as h increases.Figure 4 shows that as supermodularity increases, that is as h decreases, objective function of full infor-mation disclosure approaches to the optimal objective function. Also, full information disclosure becomesthe optimal solution for a wider range of λ values under public information structures as supermodularityincreases.We observe that except for the extreme values of the constant λ , there exist private signal structures thatperform better than no and full information disclosure. When λ approaches to zero, objective value of fullinformation disclosure converges to optimal value under general information structures. When λ approachesto 1, objective value of no information disclosure converges to optimal value under general informationstructures. 15 (a) h = 0 .
25 (b) h = 0 . h = 0 . h increases. (a) h = − . h = − . h = − . h ), objective function offull information disclosure approaches to the optimal objective function.16 Conclusions
We analyzed information design problem for LQG games under social welfare maximization and minimizationof action deviation objectives. We showed that full information disclosure is an optimal solution for welfaremaximization if there are common payoff states, specific submodularity or supermodularity in the game,or when we restrict the information design problem to public signals. For minimization of the deviationbetween players’ actions, we showed that no information disclosure is optimal in general. These results followthe intuition that if the objectives of the information designer and the payoffs of players are in conflict,information designer should blur or hide the information, and if their objectives align, the informationdesigner should reveal information.
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Payoff Coefficients Matrix for the Minimum Deviation BetweenPlayers’ Actions
Lemma 1 (Minimum deviation between players’ actions) . The expected value of f ( a, γ ) for minimum devi-ation between players’ actions can be written as following. E φ [ f ( a, γ )] = n (cid:88) i =1 − nn var ( a i ) + 2 n n (cid:88) i =1 n (cid:88) j =1 cov ( a i , a j ) (48) Proof. E φ [ f ( a, γ )] = − n (cid:88) i =1 E [ a i ] + 2 n n (cid:88) i =1 E [ a i n (cid:88) k =1 a k ] − n n (cid:88) i E [ a i ] − n n (cid:88) i =1 n (cid:88) j =1 E [ a i a j ] (49)= − n (cid:88) i =1 E [ a i ] + 2 n n (cid:88) i =1 E [ a i ] + 4 n n (cid:88) i =1 n (cid:88) j =1 E [ a i a j ] − n n (cid:88) i E [ a i ] − n n (cid:88) i =1 n (cid:88) j =1 E [ a i a j ] (50)= n (cid:88) i =1 − nn E [ a i ] + 2 n n (cid:88) i =1 n (cid:88) j =1 E [ a i a j ] (51)Because E [ a ] is constant, we can write (51) as below. E φ [ f ( a, γ )] = n (cid:88) i =1 − nn var ( a i ) + 2 n n (cid:88) i =1 n (cid:88) j =1 cov ( a i , a j ) (52)Note: In inner expectations, φ subscript is dropped. B Proof of Theorem 1
Proof.
We verify that the full information disclosure solution satisfies the KKT conditions. Let X ∈ P n + denote the solution in (21) under full information disclosure. Primal feasibility conditions stated in (18),(19) and (20) are satisfied by full information disclosure. We denote the associated dual variables by µ ∈ R n ( n +1) / , λ ∈ R n and Γ . Next we state the rest of the KKT conditions, i.e., dual feasibility, first orderoptimality and complementary slackness conditions respectively follow,Γ ∈ P n + , (53) F + n (cid:88) k =1 λ k R k + n (cid:88) k =1 k (cid:88) l =1 µ ( n − k + l M k,l + Γ = 0 , (54) X • Γ = 0 . (55)We check whether the above KKT conditions are satisfied by X . We start by substituting the equality forthe dual variable Γ in (54) into (55) to get the following condition, X • Γ = (cid:20) H − var ( γ )( H − ) T H − var ( γ ) var ( γ )( H − ) T var ( γ ) (cid:21) • ( (cid:20) H − I − I (cid:21) − n (cid:88) k =1 λ k R k − n (cid:88) k =1 k (cid:88) l =1 µ ( n − k + l M k,l ) = 0 . (56)20f the condition above is satisfied, then both (54) and (55) are also satisfied.We look for a uniform dual variable λ , i.e., λ k = λ, ∀ k ∈ N where λ ∈ R , that satisfies (56). DefineΞ = − (cid:80) nk =1 (cid:80) kl =1 µ ( n − k + l M k,l in matrix form. We also assume Ξ = µI, µ >
0. Thus, we have X • Γ = (cid:20) H − var ( γ )( H − ) T H − var ( γ ) var ( γ )( H − ) T var ( γ ) (cid:21) • (cid:20) (1 − λ ) H ( λ − I ( λ − I Ξ (cid:21) = 0 . (57)The dual feasibility condition in (53) requires the dual variables Γ to be positive semi-definite. We willutilize Schur complement to analyze positive definiteness of Γ. A strict version of dual feasibility conditionΓ (cid:31) (cid:31) / Ξ of block matrix Ξ of matrix Γ is positivedefinite where Γ / Ξ = (1 − λ ) H − ( λ − Iµ . (58)Thus, Γ / Ξ = (cid:40) (1 − λ ) − ( λ − /µ if i = j ; i, j ∈ N (1 − λ ) h if i (cid:54) = j ; i, j ∈ N. (59)Sum of each row of Γ / Ξ is (1 − λ ) − ( λ − /µ + ( n − − λ ) h . This is the first eigenvalue of Γ / Ξ. Restof the eigenvalues of Γ / Ξ are equal to (1 − λ )(1 + h ) − ( λ − µ . (60)Dual variable µ is the free variable in these eigenvalues. We need these eigenvalues to be positive, i.e., µ > max { ( λ − (1 − λ )(1 + h ) , } . (61)First term on the right hand side of (61) ensures that Γ / Ξ is positive definite and µ > µ satisfies (61).We can rewrite the inverse of the matrix H , i.e., H − , as follows for n ≥ H − i,j = (cid:40) ( n − h +1 − ( n − h +( n − h +1 if i = j ; i, j ∈ N h ( n − h − ( n − h − if i (cid:54) = j ; i, j ∈ N. (62)Using (62), we write out the Frobenius product terms within (57) corresponding to each of the four sub-matrices. First Frobenius product in (57) can be written as following by using the distributive property ofFrobenius inner product over matrix multiplication and substituting (62) for H − , we have( H − var ( θ )( H − ) T ) • ((1 − λ ) H ) = (cid:18) (1 − λ )( n ((2 − n ) h −
1) + ( n − n ) h )( n − h − ( n − h − (cid:19) ∗ ( n (cid:88) i =1 var ( γ i ) + 2 h n (cid:88) i =1 n (cid:88) j =1 cov ( γ i , γ j )) . (63)21econd Frobenius product corresponding to the off-diagonal elements can be written as( H − var ( γ )) • (( λ − I ) = ( var ( γ )( H − ) T ) • (( λ − I ) (64)= ( λ − − n ) h − (cid:80) ni =1 var ( γ i ) + 2 h (cid:80) i (cid:54) = j cov ( γ i , γ j )]( n − h − ( n − h − . (65)The second equality (65) comes from the fact that the product of non-diagonal elements of var ( γ )( H − ) T with the corresponding elements of the sub-matrix ( λ − I is equal to zero. The remaining products aregiven as var ( γ ) • Ξ = µ n (cid:88) i =1 var ( γ i ) . (66)Combining the above terms, we can expand the equality (57) as follows X • Γ = n (1 − λ ) ∗ ( n (cid:88) i =1 var ( γ i ) + h (cid:88) i (cid:54) = j cov ( γ i , γ j ))+ 2 ( λ − − n ) h − (cid:80) ni =1 var ( γ i ) + 2 h (cid:80) i (cid:54) = j cov ( γ i , γ j )]( n − h − ( n − h − µ n (cid:88) i =1 var ( γ i ) = 0 . (67)Next we show that there exists at least one real root of (67) with respect to λ with µ satisfying (61).If there is such a real root, there exists a λ ∈ R satisfying the KKT conditions. Let τ = (cid:80) ni =1 var ( γ i ) and φ = 2 (cid:80) i (cid:54) = j cov ( γ i , γ j ) to simplify the exposition.We first consider the case µ = ( λ − (1 − λ )(1+ h ) + (cid:15), (cid:15) >
0. In this case, (67) becomes X • Γ = n (1 − λ ) ( τ + hφ ) + 2( λ − − n ) h − τ + hφ ]( n − h − ( n − h − λ − (1 − λ )(1 + h ) + (cid:15) ) τ = 0 . (68)When we equalize denominators, (68) becomes a cubic equation. The cubic equation with real coefficientsalways has at least one real root. Secondly, we consider the case µ = (cid:15), (cid:15) >
0. In this case, (67) becomes X • Γ = n (1 − λ ) ∗ ( τ + hφ ) + 2 ( λ − − n ) h − τ + hφ ]( n − h − ( n − h − (cid:15)τ = 0 , (69)where a , b and c are defined as a = n ( τ + hφ ) (70) b = − n ( τ + hφ ) + 2 ((2 − n ) h − τ + hφ ( n − h + (2 − n ) h − c = n ( τ + hφ ) + − − n ) h − τ − hφ ( n − h + (2 − n ) h − (cid:15)τ (72)We want to show b − ac >
0, so that there exists a root λ r > λ r will satisfy (61). It can be easilyobserved that ( n − h − ( n − h − < < h < . Also, by our assumption τ ≥ hφ . Thus, we can22educe that the discriminant is positive, i.e., b − ac = (cid:18) − n ( τ + hφ ) + 2 ((2 − n ) h − τ + hφ ( n − h + (2 − n ) h − (cid:19) − n ( τ + hφ ) − n ( τ + hφ )[ − − n ) h − τ − hφ ]( n − h + (2 − n ) h − n ( τ + hφ ) (cid:15)τ = 8 n ( τ + hφ )[((2 − n ) h − τ + hφ ]( n − h + (2 − n ) h − (cid:18) ((2 − n ) h − τ + hφ ( n − h + (2 − n ) h − (cid:19) + n ( τ + hφ ) (cid:15)τ > . Therefore roots of (69) are real. We also need to show at least one of roots of (69) λ r is such that λ r > λ r = 1 − ((2 − n ) h − τ + hφn ( τ + hφ )[( n − h + (2 − n ) h −
1] + √ b − ac a > . (73)We know a >
0. Also, it can be deduced that the third term (73) is greater than the absolute value of thesecond term in (73) from prior discussion. Thus, λ r > ..