How Peer Effects Influence Energy Consumption
Datong P. Zhou, Mardavij Roozbehani, Munther A. Dahleh, Claire J. Tomlin
HHow Peer Effects Influence Energy Consumption
Datong P. Zhou ∗† , Mardavij Roozbehani † , Munther A. Dahleh † , and Claire J. Tomlin (cid:63) Abstract — This paper analyzes the impact of peer effectson electricity consumption of a network of rational, utility-maximizing users. Users derive utility from consuming elec-tricity as well as consuming less energy than their neighbors.However, a disutility is incurred for consuming more than theirneighbors. To maximize the profit of the load-serving entity thatprovides electricity to such users, we develop a two-stage game-theoretic model, where the entity sets the prices in the firststage. In the second stage, consumers decide on their demandin response to the observed price set in the first stage so asto maximize their utility. To this end, we derive theoreticalstatements under which such peer effects reduce aggregate userconsumption. Further, we obtain expressions for the resultingelectricity consumption and profit of the load serving entity forthe case of perfect price discrimination and a single price undercomplete information, and approximations under incompleteinformation. Simulations suggest that exposing only a selectedsubset of all users to peer effects maximizes the entity’s profit.
I. I
NTRODUCTION
Energy efficiency programs have emerged as a viableresource to yield economic benefits to utility systems and toreduce the amount of greenhouse gas emissions. Demand-side management aims to modify consumer demand throughfinancial incentive schemes and to induce behavioral changesthrough education. Specifically, users are offered rewardsto conserve energy during peak hours or to shift usageto off-peak times. With communications and informationtechnology constantly improving, which are characteristicelements of today’s smart grid, demand-side managementtechnologies are becoming increasingly feasible.Previous academic work by psychologists, political scien-tists, and behavioral economists has found that social com-parisons can have a significant impact on people’s behavior,exploiting the willingness of individuals to conform to astandard, receive social acclaim, or simply the belief thatother people’s choices are informative in the presence oflimited or imperfect information [1], [2].Motivated by this line of academic work and the pressingneed to improve energy efficiency, various companies andgroups, for instance
OPOWER , have conducted randomizedcontrol trials to investigate the impact of peer effects onenergy consumption of residential households by sending outquarterly energy reports (so called
Home Energy Reports ) ∗ Department of Mechanical Engineering, University of California, Berke-ley, USA. [email protected] † Laboratory for Information and Decision Systems, MIT, Cambridge,USA. [datong,mardavij,dahleh]@mit.edu (cid:63)
Department of Electrical Engineering and Computer Sciences, Univer-sity of California, Berkeley, USA. [email protected]
This work has been supported in part by the National Science Foundationunder CPS:FORCES (CNS-1239166) and CEC Grant 15-083. to users with a comparison of their usage to their closestneighbors [3]. While all experiments unanimously found anaverage reduction among the highest consuming users ofaround 1-2 % [4], ambiguous results were found among lowconsumers, with one study reporting a “boomerang effect”,that is, an increase of energy demand among the mostefficient households.Network effects in social networks and platforms oftenexhibit positive externalities, capturing the intuitive factthat an increased amount of platform activity promotes alocal increase in platform activity. From a game-theoreticperspective, it is known that an analysis of games undersuch strategic complements admits well-behaved solutionsif utility functions are supermodular with parameters drawnfrom a lattice [5], [6]. Examples for such games can be foundin modeling technology adoption, human capital decisions,and criminal and social networks [7]. The opposite effect,that is, in games of strategic substitutes where an increasedamount of activity leads to local reductions of activity, isobserved in information sharing and the provision of publicgoods [8]. However, since utility functions in this setting tendto lose the feature of supermodularity, finding equilibria isan inherently hard problem [9], and so these settings havebeen significantly less studied.In an attempt to characterize the most influential players ina network, [10] develops a quadratic model with continuousaction spaces, a parameterization which we employ in thispaper. Other research directions aiming at understandingthe impact of network effects on social phenomena includediffusion models for the spread of information with the goalof influence maximization [11], repeated games to learnuser interactions over time [12], or the analysis of systemicrisk and stability [13] in financial networks. The problemof profit maximization of a monopolist selling a divisiblegood, which is closely related with our work, has beeninvestigated in [14], where the authors assume a constantmarginal cost of production. However, to the best of ourknowledge, a modeling approach for the impact of peereffects on energy consumption, whose generation typicallyhas quadratic marginal cost, has yet to be formulated.In this paper, we propose a two-stage game-theoreticmodel for the energy consumption of a network of users,serviced by the load-serving entity that is obligated to coverthe households’ energy demand at all times. We analyticallysolve for the equilibria of this game under full information ofthe network structure and users’ parameters to characterizethe influence of peer effects on aggregate consumption andutility profit, for both the case of perfect price discriminationand a single price valid for all users. For the case of a r X i v : . [ c s . S I] M a r ncomplete information, we obtain approximations of theutility’s profit, user consumptions, and the optimal pricingscheme. Further, we analyze the profit-maximization problemby selecting the best subset of users to be exposed to peereffects, and present a heuristic solution to this NP-hardselection problem. Lastly, we provide theoretical statementson the properties of users which ensure that the consumptionunder peer effects is reduced.The remainder of this paper is organized as follows:Section II presents the two-stage game-theoretic model be-tween the utility and the network of consumers and derivesconsumption and price equilibria. Based on this model,Section III presents various theorems on the reduction ofconsumption in response to the peer effect as well as on theeffect of uncertainty of the network structure on the optimalprofit. Section IV compares the utility’s profit under thepricing schemes derived in Section II. Next, the challengeof maximizing the utility’s profit by imposing a binaryconstraint on the number of users exposed to peer effects isformulated and solved with a heuristic approach in SectionV. Section VI concludes the paper. All proofs are relegatedto the Appendix.II. G AME -T HEORETIC M ODEL
A. Players
Define the set of consumers as I = { , . . . , n } . Let W ∈ R n × n define the interaction matrix which describes thenetwork links and strengths between users. More precisely,let w ij ∈ [0 , denote the strength of influence of user j on i . We assume w ii = 0 ∀ i ∈ I and normalize the rowsums, (cid:80) j ∈I w ij = 1 ∀ i ∈ I . Each element w ij > in W corresponds to a directed edge from agent j to agent i , thatis, the adjacency matrix G of the resulting directed graphis the transpose of W . Each user i derives a utility u i ∈ R from consuming x i units of electricity as follows: u i = a i x i − b i x i − p i x i + γ i x i (cid:88) j ∈I w ij x j − x i . (1)In (1), a i and b i denote user-specific parameters to describethe concave and increasing direct utility from consuming x i units of electricity, and p i denotes the unit price set by theutility. The last term captures the strategic complementaritybetween user i and its neighbors. It is positive if user i consumes less than the average of its neighbors, and viceversa. The difference between the average consumption andthe user consumption is scaled by a proportionality constant γ i and the consumption level x i .Since each user consumes x i units of electricity at unitprice p i , the utility’s profit reads as follows: Π = (cid:88) i ∈I p i x i − c i x i , (2)where the marginal cost of production c i x i is assumed tobe linear in the production quantity x i , which is a standardand often made assumption. For expositional ease, we furtherassume that the utility generates electricity itself and does not procure it from the wholesale electricity market. Relaxingthis assumption would introduce uncertainty in wholesaleprices, a problem which is outside the scope of this paper. B. Two-Stage Game
To model the hierarchy between the utility, which acts asa monopolist that has the power to set prices, and the users,we formulate a two-stage game as follows:1) The utility determines the optimal price p ∗ so asto maximize its profit by taking into account users’consumption decisions as a function of any particularprice vector p , that is, p ∗ = arg max p ≥ (cid:88) i ∈I p i x i ( p i ) − c i x i ( p i ) (3)2) Each agent observes the price p ∗ i and x − i and con-sumes x ∗ i units of electricity so as to maximize herutility, that is, x ∗ i = arg max x i ≥ u i ( x i , x − i , γ i , W ) .We will solve this two-stage game by finding a subgameperfect equilibrium for the cases of perfect price discrimina-tion and a single price for all users. We also differentiatebetween the full-information case where the utility hasknowledge about all { a i } ni =1 and { b i } ni =1 , and the case inwhich only their expectations E [ a ] and E [ b ] are known. C. Subgame-Perfect Equilibrium
Assumption 1. a i > p i and b i > γ i ∀ i ∈ I . Theorem 1.
Given the price vector p and consumptionvector x − i , the utility maximizing response of user i is x ∗ i = a i − p i + γ i (cid:80) j ∈I w ij x j b i + γ i ) . (4) Further, { x ∗ , . . . , x ∗ n } constitute a unique Nash Equilibriumof the second stage game. Recall that w ii = 0 ∀ i ∈ I , which allows the right handside of (4) to depend on x − i only. Assumption 1 is necessaryto ensure that (4) is indeed a maximum attained at a non-negative value. With the definitions B := diag (2 b , . . . , b n ) and Γ := diag ( γ , . . . , γ n ) , (4) can be rewritten as x ∗ = ( B + 2Γ − Γ W ) − ( a − p ) . (5) Definition 1 (Katz-Bonacich Centrality [15], [16]) . Giventhe adjacency matrix G , the weight vector w , and the scalar ≤ α < /ρ ( G ) , where ρ ( G ) denotes the spectral radius of G , the weighted Katz-Bonacich Centrality is defined as K w ( G, α ) = ( I − αG ) − w = ∞ (cid:88) k =0 ( αG ) k w . (6) The centrality of a particular node i can be interpreted asthe sum of total number of walks from i to its neighborsdiscounted exponentially by α and weighted by w i . For the special case γ = . . . = γ n = γ , and noting that G = W (cid:62) , (5) can be rewritten in terms of the weightedatz-Bonacich Centrality: x ∗ = ( B + 2 γI ) − (cid:0) I − γW (cid:62) ( B + 2 γI ) − (cid:1) − ( a − p )= ( B + 2 γI ) − K a − p ( W (cid:62) ( B + 2 γI ) − , γ ) We note that ( B + 2Γ − Γ W ) is strictly diagonally dominantfor all γ ≥ , with positive diagonal entries. The GershgorinCircle Theorem then states that all its eigenvalues are strictlypositive, from which invertibility follows.We first focus on the full information case and present theequilibria in Theorems 2 and 3. Let C = diag ( c , . . . , c n ) . Theorem 2.
Under perfect price discrimination, the profit-maximizing solution p ∗ to the first stage game is p ∗ = a (cid:124)(cid:123)(cid:122)(cid:125) (1) + CZ a (cid:124) (cid:123)(cid:122) (cid:125) (2) − W (cid:62) Γ Z a (cid:124) (cid:123)(cid:122) (cid:125) (3) + Γ W Z a (cid:124) (cid:123)(cid:122) (cid:125) (4) , (7) Z = (cid:20)
2Γ + B + C − (cid:18) W (cid:62) Γ2 + Γ W (cid:19)(cid:21) − . The four components are interpreted as follows: A constant term a i / , c.f. a i in (1) , An additional cost that correlates with cost c i , An incentive for strongly influential users W (cid:62) Γ , An additional cost for strongly influenced users Γ W .The optimal consumption under this policy is x ∗ = (cid:18) C + B + 2Γ − W (cid:62) Γ2 − Γ W (cid:19) − a . (8) For the special case of symmetric networks, i.e. W = W (cid:62) ,the optimal profit Π ∗ becomes Π ∗ = 14 a (cid:62) ( C + B + 2Γ − Γ W ) − a . (9) Theorem 3.
Under complete information, i.e. the utilityknows a i and b i ∀ i ∈ I , the profit-maximizing single price p ∗ u is p ∗ u = (cid:20) − (cid:62) A − · (cid:62) ( A − + A − CA − ) (cid:21) ¯ a (10) and the consumption equilibrium writes x ∗ = A − (cid:20) a − (cid:18) − (cid:62) A − · (cid:62) ( A − + A − CA − ) (cid:19) ¯ a (cid:21) , (11) where A = B + 2Γ − Γ W and ¯ a = (cid:80) ni =1 a i /n . Lemma 1.
For symmetric networks, i.e. W = W (cid:62) , thesingle profit-maximizing price (10) and its correspondingconsumption (11) simplify to p ∗ u = 1 n n (cid:88) i =1 p ∗ i (12a) x ∗ u = ( B + 2Γ − Γ W ) − ( a − ¯ a ) (12b) + ( C + B + 2Γ − Γ W ) − ¯ a . By construction of the optimal prices and consumptions,the optimal profit under a single price is less than underperfect price discrimination, that is, Π ∗ u ≥ Π ∗ .Next, for the incomplete information scenario and addi-tional assumptions W = W (cid:62) and C = cI , the utility canapproximate the profit-maximizing price as in Theorem 4. Theorem 4.
In the case of incomplete information, that is,only the expectations of { a i } ni =1 and { b i } ni =1 are knownand denoted with E [ a ] and E [ b ] , the optimal single profit-maximizing price ˜ p ∗ u and the expected corresponding con-sumption equilibrium E [˜ x i ] are bounded below by ˜ p ∗ u ≥ E [ a ]2 (cid:104) cn (cid:62) [2Γ + (2 E [ b ] + c ) I − Γ W ] − (cid:105) , (13a) E [˜ x i ] ≥ E [ a ] − ˜ p ∗ u, LB n · (cid:62) (2Γ + 2 E [ b ] I − Γ W ) − . (13b) where ˜ p ∗ u, LB denotes the lower bound on the single profit-maximizing price ˜ p ∗ u (13a) . Theorem 5 (Profit Maximizing Price without Peer Effects) . In the case of incomplete information and in the absence ofany peer effects, the single profit-maximizing price ˆ p ∗ andthe expected user consumption E [ˆ x i ] are ˆ p ∗ = E [ b ] + c E [ b ] + c E [ a ] , (14a) E [ˆ x ∗ i ] = E [ a ]2(2 E [ b ] + c ) ∀ i ∈ I . (14b)III. T HEORETICAL S TATEMENTS
We next seek to analyze under what conditions the aggre-gate consumption across all users is less than in the absenceof peer effects, which is a desirable goal from the energyefficiency perspective.
Theorem 6. If a i =: a , b i =: b , and γ i =: γ ∀ i ∈ I , andAssumption 1 holds, then x ∗ i (4) is strictly monotonicallydecreasing in γ , independent of the network topology W . Theorem 6 is interesting because identical consumers willreduce their optimal consumption compared to the case ofno peer effects, even though x ∗ i = x ∗ j ∀ i, j ∈ I and hencethe peer effect term γ i x i (cid:16)(cid:80) j ∈I w ij x j − x i (cid:17) is zero. Theorem 7 ( Influence of High Consumer ) . Given that w ij = (cid:16)(cid:80) j ∈I w ij > (cid:17) − ∀ i ∈ I , that is, all connectionsare of equal weight, and b i =: b and γ i =: γ ∀ i ∈ I .Define the set of users N := { i ∈ I \ j } with thecharacteristic a i − p i =: α ∀ i ∈ N . Further, let j be a“high consumer”, that is, a j − p j =: ¯ α > nα . Denote theset of all neighbors of j as C j := { i ∈ N | w ij > } . Then,independent of the network topology, for all users i ∈ C j , x ∗ i is increasing for small enough values of γ whereas x ∗ j isstrictly monotonically decreasing in γ . Let m i denote the number of neighbors of consumer i .Theorem 7 can be restated as in Lemma 2. emma 2. x ∗ i , i ∈ C j is increasing for small enough valuesof γ if ¯ α ≥ m j + 1 . Equivalently, if ¯ α = kα, k ∈ N , onlythe subset { i ∈ C j | m i ≤ k − } , i.e. the set of userswith fewer than k − neighbors, shows an initial increasein consumption as a function of γ . Theorem 7 and Lemma 2 describe conditions on theaverage consumption of any particular user’s neighbors toobserve a “boomerang effect”, given there is a unique “high”consumer among a pool of users of identical characteristics.
Theorem 8 ( Targeted Peer Effects ) . For a general settingof n ≥ users with non-identical parameters a i , b i and afixed price p among all users, exposing exactly two connectedusers to the peer effect, w.l.o.g. referred to as users “1”and “2”, reduces the sum of their consumptions under thefollowing conditions: b ≤ ( a − p ) [4( b + γ ) − γw w ]4( b + γ ) n (cid:80) j =3 w j x j + 2 w (cid:18) a − p + γ n (cid:80) j =3 w j x j (cid:19) (15a) b ≤ ( a − p ) [4( b + γ ) − γw w ]4( b + γ ) n (cid:80) j =3 w j x j + 2 w (cid:18) a − p + γ n (cid:80) j =3 w j x j (cid:19) (15b) where x j , j ∈ { , . . . , n } is given by x j = ( a j − p ) / (2 b j ) .For the special case of n = 2 , this condition reads b ≤ ( a − p ) (4 b + 3 γ )2( a − p ) and b ≤ ( a − p ) (4 b + 3 γ )2( a − p ) Theorem 8 states that if two connected users both receivenotifications of their neighbors’ consumption, the sum oftheir consumptions decreases as long as they are not “toodifferent” from each other and their neighbors. Thus, the totalconsumption of a network of users correlates negatively withthe number of users given the treatment. Analogous boundscan be found for exposing more than two users to the peereffect at the expense of notational ease.Finally, we investigate the case of incomplete informa-tion about the network structure for the case of symmetricnetworks, i.e. W = W (cid:62) . It is assumed that the monopolistonly knows an approximation of W , denoted with ˜ W , where ˜ W = ˜ W (cid:62) . Under perfect price discrimination, the utilitycan set profit-maximizing prices in the first stage of thegame, assuming that users’ consumption ˜x in the first stageis determined according to ˜ W . The real consumption x ∗ ,however, follows the actual W (which is unknown to theutility). Theorem 9 provides a lower bound on the ratio ofthe optimal expected profit under network uncertainty to theprofit obtainable under perfect network information. Theorem 9 ( Uncertainty in W ) . Assume that W = W (cid:62) and Γ = γI, γ ≥ . If the monopolist has access only tothe estimate ˜ W with ˜ W = ˜ W (cid:62) , then, under perfect pricediscrimination, the ratio of optimal expected profit ˜Π ∗ to profit Π ∗ under perfect knowledge of W is bounded below: ˜Π ∗ Π ∗ ≥ λ min ( C + B + 2Γ − Γ W ) λ max ( C + B + 2Γ − Γ W ) + γ (cid:107) W − ˜ W (cid:107) , (16) where (cid:107) · (cid:107) is the Euclidian matrix norm. For the edge case ˜ W = 0 , we have (cid:107) W (cid:107) = 1 due to thewell-known fact that the maximal eigenvalue of an adjacencymatrix is the degree of the graph. Due to row normalizationsof W , the degree is , which corresponds to the eigenvector associated with eigenvalue 1. To qualitatively show that thebound (16) becomes tighter as ˜ W approaches W , observethat (cid:107) W − ˜ W (cid:107) corresponds to the largest singular valueof W − ˜ W , which is identical to its spectral radius because W − ˜ W is Hermitian. Finally, the Gershgorin Circle Theoremstates that every eigenvalue of W − ˜ W lies within at leastone of the disks that is centered at the origin, each of whichhas radius R i = (cid:80) j (cid:54) = i | w ij − ˜ w ij | . As w ij → ˜ w ij , R i → .To illustrate the bound (16), let n = 24 and W ∈ R × be the ground truth interaction matrix of 12 randomly chosen,fully connected users, whose parameters a i , b i , and c i ∀ i ∈ I are randomly drawn from appropriate distributions. Assum-ing that the monopolist knows that 12 out of 24 users arefully connected, we iterate through all (cid:0) (cid:1) combinations andcalculate (cid:107) ˜ W − W (cid:107) and the profit bound (16) as a functionof the number of correct user assignments, where we take themean across any particular number of correct assignments.As the number of correct assignments increases, the metricfor the mismatch between W and ˜ W , namely (cid:107) ˜ W − W (cid:107) decreases, whereas the profit bound increases, see Figure 1. Number of Correct Guesses . . . . . . k ˜ W − W k Uncertainty of Interaction Matrix
Number of Correct Guesses . . . . . . ˜ Π ∗ / Π ∗ Bounds on Monopolist Profit
Profit RatioLower Bound
Fig. 1: (cid:107) W − ˜ W (cid:107) for 12 fully connected users embedded in a network of n = 24 customers, γ = 0 . . Theorem 10 (Efficiency) . The consumption equilibrium x ∗ (8) is inefficient as the social welfare S attained at (8) issuboptimal. Specifically, x ∗ i < x oi ∀ i ∈ I , where x o denotesthe consumption that maximizes social welfare, which reads x o = (cid:18) C + B − W (cid:62) Γ2 − Γ W (cid:19) − a . (17) Allocating users per-unit subsidies s i = ( b i + γ i ) x i / (Pigouvian Subsidy) can restore the social optimum. IV. C
OMPARISON OF P RICING S CHEMES
A. Network Topologies
In the remainder of this paper, we assume users to beconnected to each other through one of the basic networktopologies displayed in Figure 2.3 42 13 42 13 42
Fig. 2: Basic network architectures for n = 4 : Fully connected, star, ring B. Simulation
We now simulate the consumption and price equilibriaas well as the profit of the monopolist as a function ofthe network strength parameter γ under the following threepricing scenarios: • Case 1 : Monopolist has complete information of a and b and sets prices with perfect price discrimination (7); • Case 2 : Monopolist has complete information of a and b and sets the profit-maximizing single price (10); • Case 3 : Monopolist has access only to E [ a ] and E [ b ] and sets the lower bound on the single price (13a).We simulate a network of n = 10 fully connected userswith a i and b i randomly drawn from uniform distributionswith support [8 , and [0 . , . , respectively. The costis set to c i = 2 for all users. As the results for the star andring network are qualitatively similar to the fully connectednetwork, we omit discussions of these cases. The optimalprices for each of the cases (1)-(3) are then calculated, whichfixes the users’ consumptions and the monopolist’s profit.Repeating this process 10,000 times and taking the meanacross all iterations yields the characteristics in Figure 3.As expected, the profit under perfect price discrimination(7) exceeds the profit obtained with cases (2) and (3), where,somewhat surprisingly, setting the lower bound on the prices(case (3)) does not give up too much profit, compared tocase (2). This indicates that the lower bound on the optimalprice (13a) is “close” to the actual optimum, which is provenby the second subplot, from which it follows that (13a) fallsshort of (10) by less than < .Consequently, the lower price bound (13a) results in ahigher average user consumption than in case (2), whichdirectly follows from the consumption equilibrium (4). Theaverage user consumption under perfect price discriminationis sandwiched between cases (2) and (3).Lastly, the maximum user consumption for perfect pricediscrimination is about lower than in cases (2) and(3), which has beneficial side-effects on grid operation.This observation also motivates the heuristic user-selectionalgorithm presented in the next section.V. P ROFIT M AXIMIZATION WITH U SER S ELECTION
A. Problem Formulation
We now seek to answer the following question: Giventhe single, exogenous price p and the parameters { a i } ni =1 and { b i } ni =1 sampled from distributions with means E [ a ] and E [ b ] , respectively and are known to the monopolist, whichusers should be targeted to maximize profit? This situationcan arise if the utility is obligated to charge customers at a .
00 0 .
05 0 .
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25 0 . Monopolist Profit Π .
00 0 .
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25 0 . . . . Single Price for Cases 2 and 3 .
00 0 .
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Perfect Price DiscriminationComplete Information, Single PriceIncomplete Information, Single Price Bound .
00 0 .
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25 0 . Interaction strength γ . . . . Maximum User Consumption
Fig. 3: Profit of monopolist, single prices (10) and (13a), average userconsumption, and maximum consumption under perfect price discrimination(green), single pricing under complete information (10) (yellow), and singlepricing under incomplete information (13a) (red). 10,000 iterations, a ∼ unif [8 , , b ∼ unif [0 . , . , c i = 2 ∀ i ∈ I . rate p per unit of electricity and only wants to spend a limitedbudget on informing users about their peers’ behavior. Inother words, which best subset of all users should be exposedto the peer effect such that the utility achieves maximumprofit under exogenous price p ? The profit maximizingproblem of the utility thus writesmaximize δ ,...,δ n n (cid:88) i =1 px i − c i x i subject to x = ( B + 2∆Γ − ∆Γ W ) − ( a − p ) n (cid:88) i =1 δ i = m, δ i ∈ { , } (18)where ∆ = diag ( δ , . . . , δ n ) and δ i = 1 and δ i = 0 denote that user i is targeted or non-targeted, respectively.This is an NP-hard Mixed Integer Quadratically ConstrainedProgram (MIQCP) due to the binary constraint to exposeexactly m , ≤ m ≤ n users to the network effect and thequadratic objective, and so (18) does not admit a closed formsolution. An analytical solution requires exhaustive search,which is computationally infeasible for any real network ofusers. Therefore, we resort to the following heuristic whichwas hinted at at the end of Section IV: Given the userparameters a and b and the single price p , we first computethe consumptions in the absence of any network effects,denoted with ˜ x = B − ( a − p ) . Next, we calculate theoptimal consumptions with the expectations of E [ a ] and E [ b ] ,which we denote with E [ x ] . Lastly, the pairwise differences E [ x ] − ˜ x i | are put into a sorted list, and the heuristic selectionalgorithm returns the indices of the m largest values in thislist. That is, ∆ h = diag ( δ h, , . . . , δ h,n ) , where δ h,i = 1 ifconsumer i belongs to the set of the m largest | E [ x ] − ˜ x i | ,and δ h,i = 0 otherwise.The idea of this heuristic is motivated by Theorem 7,according to which a high consumer in a network of lowconsumers can result in a consumption increase of lowconsumers. Since the user parameters are sampled froma finite distribution, a single price on non-identical usersalways results in suboptimal profit, but approaches optimalityas users become more similar. Exposing the highest andlowest consumers (measured against E [ x ] ) to the networkeffect nudges high users (low users) to consume less (more),thereby making the users more similar in their consumption,which in turn increases the utility’s profit.Further, the fact that the maximum user consumptionunder perfect price discrimination (which achieves notablybetter profit than single pricing, see Figure 3) is about 30 %lower than under single pricing corroborates the notion ofexposing high consumers to the peer effects. According toTheorem 7, such users reduce their consumption in responseto the peer effect, which reduces the maximum user con-sumption to increase profit.The utility needs to find the sweet spot between thefollowing two extremes: Targeting too few users results in asuboptimal increase in profit. On the other hand, accordingto Theorem 8, targeting too many users leads to an overallconsumption decrease because targeting a customer whoseneighbors are already exposed to the network causes theneighbors to reduce their consumption further.Note that this heuristic neither takes into account theinteraction matrix W nor the fact whether the deviation ofthe actual consumption from the expected one is positive ornegative, and so it could be improved by running a classifi-cation algorithm on the features | E [ x ] − ˜ x i | + , | E [ x ] − ˜ x i | − ,and γ i ˜ x i (cid:16)(cid:80) j ∈I w ij ˜ x j − ˜ x i (cid:17) . B. Simulation
We let c i = 2 , n = 10 as in Section IV and analyzeall three network topologies depicted in Figure 2. a i and b i are sampled from the same uniform distributions. Weset the exogenous price as the profit-maximizing price inthe absence of peer effects (14a), from which the expectedconsumption E [ x ] is determined with (14b). The analyticalsolution to the MIQCP (18) is determined with Gurobi [17]. We repeat this calculation 10,000 times and take themean across all iterations. To describe the performance ofthe heuristic, we define the performance metric S as follows: S m = Π hm − Π E Π ∗ m − Π E · , (19)where Π ∗ m and Π hm denote the profit under the analyticalsolution of (18) and the heuristic with m targeted users,respectively. Π E denotes the profit in the absence of any peereffects ( m = 0 ) achieved with exogenous price p where theusers consume according to ˜ x = B − ( a − p ) . S m captures the fraction of the heuristic’s achieved profit improvement ofthe total possible improvement.Figure 4 shows the objective for the heuristic Π h (solidlines) and analytical solution Π ∗ (colored dashed line) for allnetwork topologies as a function of m . The expected profitwith m = 0 follows by taking the expectation of the profit E [Π] m =0 = n · E a ∼ U [8 , E b ∼ U [0 . , . (cid:2) px − cx (cid:3) (cid:12)(cid:12)(cid:12) x = a − p b , which is depicted as the black dashed line. Further, thepercentage of cases where the heuristic selects the identicalsubset of users as the analytical solution is depicted in thesecond subplot. S m and the maximum user consumption asa function of m are provided in the third and fourth subplot,respectively. . . . E [Π] m =0 Monopolist Profit, Analytical Π ∗ (dashed) vs. Heuristic Π h (solid) Percentage of Optimal Choices by Heuristic
Performance S [ % ] of Heuristic Number Targeted . . . Maximum Consumption under Heuristic
Fully ConnectedRing NetworkStar Network
Fig. 4: Average profit, percentage of optimal choice of heuristic, regret, andaverage infinity norm of consumption for the utility’s profit maximizationproblem under the single price (10). 10,000 iterations, a ∼ unif [8 , , b ∼ unif [0 . , . , c i = 2 . For all network topologies, it can be seen that the optimalsolution to (18) achieves an increase in profit by ≈ for m ∈ { , , } compared to the case of no targeting,while at the same time reducing the peak consumption by ≈ . The performance of the heuristic decreases in thenumber of consumers targeted and reaches its minimum at ≈ , ≈ , and ≈ for the ring, star, and fullyconnected network, respectively. The percentage of optimalchoices across all 10,000 iterations is always > . Theseresults suggest that the presented heuristic achieves a goodapproximation of the optimal solution, which is NP-hard andcomputationally intractable for larger, real-world networks.I. C ONCLUSION
Motivated by home energy reports that benchmark theconsumption of individual users against their neighbors, weproposed a two-stage game-theoretic model for a networkof electricity consumers, in which each consumer seeks tooptimize her individual utility function that includes a peereffect term. Specifically, users derive positive utility fromconsuming less energy than the average of their neighbors,and vice versa. We investigated profit-maximizing pricingschemes for the complete and incomplete information sce-nario as well as for the single price and perfect pricediscrimination case. We provided theoretical statements withregard to overall consumption, efficiency, and profit undernetwork uncertainty. For the case of targeting only a sub-set of all available consumers under an exogenous singleprice, we formulated the monopolist’s profit maximizationproblem. The resulting NP-hard optimization problem wassolved with a heuristic approach, which simply targets thoseusers who deviate most from the expected consumptionin the hypothetical absence of peer effects. Compared tothe analytical solution, this heuristic was shown to achieveacceptable accuracy.This work could be extended by incorporating time. In par-ticular, if we allow the monopolist to also procure electricityfrom the wholesale market whose prices are fluctuating, analgorithmic and online treatment of this problem becomesnecessary. The goal then becomes to learn user preferencesand the network structure over time. Further, the selectionproblem to target the most valuable users for the objective ofprofit maximization calls for modeling peer effects in auctionsettings, where the desired goal is to design a truthful andincentive compatible mechanism to elicit user preferences.R
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Proof of Theorem 1
With Assumption 1, (4) follows by evaluating the firstorder optimality condition of (1) and acknowledging that itssecond derivative is strictly negative. Uniqueness of the NashEquilibrium follows from Topkis’ Theorem on supermodulargames [5], which holds due to the continuity of the payofffunctions (1) on the compact set R + and increasing differ-ences in ( x i , x − i ) as ∂ u i ∂x i ∂x j ≥ ∀ i, j ∈ I . Proof of Theorem 2 (7) is obtained by solvingmaximize p p (cid:62) x − x (cid:62) C x subject to x = ( B + 2Γ − Γ W ) − ( a − p ) (20)and applying the Matrix Inversion Lemma for general ma-trices A, U, C, V of appropriate dimensions: ( A + U CV ) − = A − − A − U (cid:0) C − + V A − U (cid:1) − V A − . The optimal profit Π ∗ is obtained by plugging p ∗ and x ∗ into the utility function of the monopolist. Proof of Theorem 3 (10) is obtained in the same fashion as (7):maximize p p (cid:62) x − x (cid:62) C x subject to x = ( B + 2Γ − Γ W ) − ( a − p ) Eliminating x from both equations and evaluating the firstorder optimality condition with respect to p yields (10). Proof of Theorem 4
To derive (13a), we first note that since W = W (cid:62) and C = cI , the profit maximizing solution under completeinformation (7) simplifies to p ∗ u = (cid:62) a n + (cid:62) [2Γ + B + cI − Γ W ] − a c n . After taking the expectation with respect to the randomvariables { a i } ni =1 and { b i } ni =1 to obtain ˜ p ∗ u = E [ a ]2 + E [ a ] c n E b (cid:104) (cid:62) [2Γ + B + cI − Γ W ] − (cid:62) (cid:105) , e first show convexity of the last term indiag (2 b , . . . , b n ) . Define the matrices D = Γ + c I − Γ W α diag (2 b , . . . , b n ) ,E = Γ + c I − Γ W − α ) diag (cid:0) b , . . . , b n (cid:1) , where α ∈ (0 , . D and E are clearly positive definite due tothe Levy-Desplanques Theorem [18]. It is then to be shownthat g ( X ) := (cid:62) X − , X := ( αD + (1 − α ) E ) − ,X := 2Γ + B + cI − Γ W is a convex function on the domain of all positive definitematrices. Using the Schur Decomposition, which states (cid:20) S T (cid:62)
T U (cid:21) (cid:23) ⇔ S (cid:23) T (cid:62) U − T, and since positive definite matrices are convex, α (cid:20) (cid:62) D − (cid:62) D (cid:21) + (1 − α ) (cid:20) (cid:62) E − (cid:62) E (cid:21) = (cid:20) α (cid:62) D − + (1 − α ) (cid:62) E − (cid:62) αD + (1 − α ) E (cid:21) (cid:23) , This immediately shows convexity of g ( X ) : αg ( D ) + (1 − α ) g ( E ) = α (cid:62) D − + (1 − α ) (cid:62) E − ≥ (cid:62) ( αD + (1 − α ) E ) − = g ( αD + (1 − α ) E ) . Finally, applying Jensen’s inequality in the multivari-ate case on the multivariate random variable Y := diag (2 b , . . . , b n ) , we obtain E Y [ g ( X )] ≥ g ( E Y [ X ]) , from which (13a) follows directly. Proof of Theorem 5
Under the given conditions, it follows immediately that E [ x ∗ ] = . . . = E [ x ∗ n ] . With this constraint, taking the ex-pectation of (4) yields E [ x ∗ ( p )] as a function of p . Plugging E [ x ∗ ( p )] into the utility’s profit function (2) and taking theexpectation with respect to a and b allows to compute theoptimal uniform price p ∗ (14b). Next, setting p = p ∗ in E [ x ∗ ( p )] yields (14b). Proof of Theorem 6
Taking the derivative of (5) with respect to γ yields: d x dγ = − γ ( b + γ ) K − F − ( a − p ) , γ > where we used the abbreviations K := (cid:18) I − γ (2 b + 2 γ ) W (cid:19) , F := (cid:18) I + bγ (2 I − W ) − (cid:19) .K is a strictly diagonally dominant M-Matrix because it canbe expressed in the form sI − B with s = 1 and has negativeoff-diagonal elements [19]. This special property guaranteesthat its inverse exists and is strictly diagonally dominant and entrywise positive. F is strictly diagonally dominant withpositive off-diagonal entries, because (2 I − W ) − is an M-Matrix . The Levy-Desplanques Theorem [18] then impliesthat F − exists, is diagonally dominant, and possesses non-negative diagonal elements. Despite the possible negativityof its off-diagonal elements, we show that the row sums of K − F − are positive. Take, for example the i -th row sum: n (cid:88) j =1 ( K − F − ) ij = n (cid:88) j =1 n (cid:88) s =1 K − is F − sj = n (cid:88) s =1 K − is F − ss + n (cid:88) s =1 K − is n (cid:88) j =1 ,j (cid:54) = s F − sj > n (cid:88) s =1 K − is F − ss − n (cid:88) s =1 K − is F − ss = 0 . Together with a i > p i ∀ i ∈ I (see Assumption 1), thisshows that d x dγ < for γ > . Proof of Theorem 7
Define L := (2 I − W ) , which is a diagonally dominantmatrix. Evaluating d x dγ at γ = 0 yields d x dγ (cid:12)(cid:12)(cid:12) γ =0 = −
14 (2 I − W )( a − p ) = − L α , (21)where α is the column vector of all { α i | i ∈ I} . Evaluatingthis derivative for user i (cid:54) = j, i ∈ C j yields − dx i dγ (cid:12)(cid:12)(cid:12) γ =0 = L ii α + L ij ¯ α + (cid:88) k ∈I\{ i,j } L ik α = 2 α − ¯ αn − − ( n − αn − < α − nαn − − ( n − αn − . Hence we have dx i dγ (cid:12)(cid:12)(cid:12) γ =0 > . On the other hand, for the“high” consumer j , the derivative reads − dx j dγ (cid:12)(cid:12)(cid:12) γ =0 = L jj α + (cid:88) k ∈I\ j L jk α = 2 α − n − n − α > , which completes the proof. Proof of Lemma 2
The proof is similar to the one used for Theorem 7. Foreach user i , i ∈ I \ j , the derivative reads − dx i dγ (cid:12)(cid:12)(cid:12) γ =0 = 2 α − kαm i − m i − m i − α = α ( m i − k ) m i − ≤ ( k − − km i − α < . For user j , we have − dx j dγ (cid:12)(cid:12)(cid:12) γ =0 = 2 α − αm j − m j − m j − α > . roof of Theorem 8 From Theorem 5, any user j with index , . . . , n , giventhe price p , consumes ( a j − p ) / (2 b j ) . To find x ∗ and x ∗ , wesolve (4) for users 1 and 2: (cid:20) b + γ ) − γw − γw b + γ ) (cid:21) (cid:20) x ∗ x ∗ (cid:21) = (cid:20) a − p + γ (cid:80) nj =3 w j x j a − p + γ (cid:80) nj =3 w j x j (cid:21) Comparing x ∗ + x ∗ to the consumptions without peer effect,that is, ( a − p ) / (2 b ) + ( a − p ) / (2 b ) yields the desiredinequalities. For the special case n = 2 , note that w = w = 1 and w j , j ≥ as well as w j , j ≥ , are zero. Proof of Theorem 9
The optimal pricing vector ˜p ∗ under network uncertaintyand its corresponding consumption vector ˜x ∗ can be deter-mined by solving (20) (with W = ˜ W ) with respect to p . ˜x ∗ is then determined by plugging ˜p ∗ back into (4). Let F := B + 2Γ − Γ W , ˜ F := λ + 2Γ − Γ ˜ W . Then ˜p and ˜x are ˜p ∗ = a − ˜ F ( C + ˜ F ) − a / , ˜x ∗ = F − ˜ F ( C + ˜ F ) − a / . The optimal profit ˜Π ∗ = ˜p ∗(cid:62) ˜x ∗ − ˜x ∗(cid:62) C ˜x ∗ can then beexpressed as follows: ˜Π ∗ = 14 a (cid:62) ( C + ˜ F ) − a + O ( γ ) ≥ a (cid:62) ( C + ˜ F ) − a . Using the definition of Rayleigh quotients [18], we thusobtain the following ratio on the profit under uncertainty: ˜Π ∗ Π ∗ ≥ a (cid:62) ( C + ˜ F ) − aa (cid:62) ( C + F ) − a ≥ λ min (( C + ˜ F ) − ) λ max (( C + F ) − ) . ( C + ˜ F ) as well as ( C + F ) are symmetric positivedefinite matrices due to their diagonal dominance withnonpositive off-diagonal elements. Hence the eigenvaluesof their inverses are strictly positive. Utilizing the identity λ min ( A ) − = 1 /λ max ( A ) for any nonsingular matrix A , and (cid:107) A + B (cid:107) ≤ (cid:107) A (cid:107) + (cid:107) B (cid:107) (a fundamental property of matrixnorms), further simplifications yield ˜Π ∗ Π ∗ ≥ λ min ( C + F ) λ max ( C + ˜ F ) = λ min ( C + F ) (cid:107) C + F + γ ( W − ˜ W ) (cid:107) ≥ λ min ( C + F ) λ max ( C + F ) + γ (cid:107) ( W − ˜ W ) (cid:107) , where we used the fact that for a symmetric positivedefinite matrix A , we have (cid:107) A (cid:107) ≡ (cid:112) λ max ( A (cid:62) A ) = (cid:112) λ max ( A ) = λ max ( A ) . Proof of Theorem 10
The social welfare S is the sum of all users’ and themonopolist’s utility: S = (cid:88) i ∈I a i x i − b i x i − c i x i + γ i x i (cid:88) j ∈I w ij x j − x i . For each i ∈ I , minimizing S with respect to x i yields d S dx i = a i − b i + c i + γ i ) x i + γ i (cid:88) j ∈I w ij x j + γ i (cid:88) j ∈I w ji x i , where the last term on the right hand side signifies theexternalities user i imposes on its neighbors, but which areunaccounted for in the individual users’ utility maximization.Solving for x i and vectorizing the equation yields (17).To show that x oi > x ∗ i for γ > , it suffices to show that A = ( C + B/ − W (cid:62) Γ / − Γ W/ − is entrywise greaterthat B = ( C + B +2Γ − W (cid:62) Γ / − Γ W/ − . By performingGauss-Jordan Elimination on A and B and exploiting the factthat A and B are diagonally dominant matrices with positivevalues on the diagonal and negative off-diagonal entries, thisclaim follows.To show that a Pigouvian Subsidy of s i = ( b i + γ i ) x i restores social welfare, note that the user’s utility function u oi now reads u oi = a i x i − b i x i − p i x i + γ i x i (cid:88) j ∈I g ij x j − x i . The solution to the subgame-perfect equilibrium under thenew user utility u oi yields x oioi