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1 Pricing access to the Internet with partial information Ilaria Brunetti, Eitan Altman, Majed Haddad To cite this version: Ilaria Brunetti, Eitan Altman, Majed Haddad. Pricing access to the Internet with partial information. Valuetools, Oct 2012, Cargese, France <hal > HAL Id: hal Submitted on 24 May 2012 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

3 2 The Signalling Game Let us first consider the following scenario, which is common knowledge for both providers (which we call players). The price p 1 of player 1 is fixed and known. Player 2 can choose one of two possible levels of pricing: ζ l and ζ h. The demand d i for each provider is a function of the prices of both providers. demand function of player 2 is known to both players. The We assume that the demand of player 1 depends on some parameter θ. θ is known to player 1 but not to player 2. It can take one of two values: θ 1 and θ 2. The game is then played as follows. The value of θ is chosen by nature according to some distribution π. It equals θ r with probability π r. Player 1 observes θ and then sends a signal to player 2. It has four pure strategies for signalling: q 1 q 1, q 1 q 2, q 2 q 1 and q 2 q 2. Here, q i q j is the strategy according to which player 1 signals q i if it observes θ 1 and signals q j if it observes θ 2. Player 2 has also four pure strategies. We write them as ζ l ζ l, ζ l ζ h, ζ h ζ l and ζ h ζ h. Here, a strategy of the form ζ i ζ j means choosing ζ i if player 2 observes a signal q 1 by player 1 and choosing ζ j otherwise. The game that we study is known as a game with cheap talk ; this is a signaling game where the sender incurs no cost for his signals [1, Sec. 7]. 2.1 Transformation into a matrix game We can now represent the game as a 4 4 matrix game. Fix i (1, 2), j (1, 2), m (l, h), n (l, h). Assume that player 1 chooses q i q j and that player 2 chooses ζ m ζ n. For r = 1, 2, define ξ(r, i, j) to be i if r = 1 and j otherwise. Player 1 sends signal q ξ(r,i,j) to player 2 if θ = θ r. Define φ(r, i, j, m, n) to be the price chosen by player 2 as a result of the signal ξ(r, i, j) when it uses the strategy ζ m ζ n. φ(r, i, j, m, n) equals ζ m if ξ(r, i, j) = 1 and is otherwise ζ n. Then U 1 (q i q j ; ζ m ζ n ) = E qi q j ;ζ mζ n [p 1 d 1 (p 1, p 2 )] (1) = π r E qi q j ;ζ mζ n [p 1 d 1 (p 1, p 2 ) θ = θ r ] (2) where E qi q j ;ζ mζ n [p 1 d 1 (p 1, p 2 ) θ = θ r ] = p 1 d 1 (p 1, φ(r, i, j, m, n). Thus we can rewrite: U 1 (q i q j ; ζ m ζ n ) = p 1 2 π r d 1 (p 1, φ(r, i, j, m, n)

4 For player 2 we obtain: 2.2 The Linear Case U 2 (q i q j ; ζ m ζ n ) = E qi q j ;ζ mζ n [p 2 d 2 (p 1, p 2 )] (3) = π r φ(r, i, j, m, n) d 2 (p 1, φ(r, i, j, m, n) (4) We now consider a linear demand function: d i = d 0 i A ii p i + A ji p j (5) Assume that A is known but that d 0 2 is unknown. It can be written as a function of a parameter θ, i.e., d 0 2 = d0 2 (θ). It follows from the above considerations that the utility functions of player 1 and player 2 can be expressed as: ) U 1 (q i q j ; ζ m ζ n ) = p 1 π r (θ r A 11 p 1 + A 21 φ(r, i, j, m, n) (6) 3 Prices as signals ) π r φ(r, i, j, m, n) (d 0 2(θ r ) A 22 φ(r, i, j, m, n) + A 12 p 1 Now consider the following variant of the problem. Instead of a fixed price p 1, the first player can choose one of two values as its own price instead of just a signle. Yet the price itself serves also as a signal that player 2 can see. q i, i = 1, 2 now stands for the price chosen by player 1. In this case assuming that q i stands for the price chosen by player 1 and following the previous definitions, we have that q ξ(r,i,j) corresponds to the price chosen by player 1 if he plays strategy q i q j and θ = θ r. The utility functions we obtain are: The Linear Case π r q ξ(r,i,j) (d 1 (q ξ(r,i,j), φ(r, i, j, m, n))) π r φ(r, i, j, m, n)(d 2 (q ξ(r,i,j), φ(r, i, j, m, n))) If we consider again the linear demand function defined in (5), the utility functions of player 1 and player 2 can be expressed as: π r q ξ(r,i,j) (θ r A 11 q ξ(r,i,j) + A 21 φ(r, i, j, m, n)) π r φ(r, i, j, m, n)(d 0 2(θ r ) A 22 φ(r, i, j, m, n) + A 12 q ξ(r,i,j) ) 3 (7)

5 It follows from the above considerations that p 1 = q ξ(r,i,j), yielding the following expressions for the utility functions of player 1 and player 2 π r q ξ(r,i,j) (θ r A 11 q ξ(r,i,j) + A 21 φ(r, i, j, m, n)) (8) 4 Discussion π r φ(r, i, j, m, n)(d 0 2(θ r ) A 22 φ(r, i, j, m, n) + A 12 q ξ(r,i,j) ) (9) In this paper, we studied a model of pricing access to the Internet and investigated the role of signalling (or recommendations) in identifying side payments (price per unit demand). We considered the case of two Internet service providers aiming at maximizing their own expected utility. Under this setting, we assume that that the amount of information made available to both ISPs on the demand function is not the same: the more informed provider plays first, and then the second observes the move of the first provider and chooses accordingly its own action. We studied two configurations of the proposed problem. In the first configuration, one service provider (say provider 1) has a fixed price known to the other provider (say provider 2). The pure actions of provider 1 are thus the possible choices of what signal to send. In the second configuration, provider 1 is allowed to choose his prices which serves as a signal to provider 2. We then reduced each one of the configuration to an equivalent four by four matrix game and studied the equilibria Our model is, needless to say, a mere caricature that captures certain types of interactions between ISPs. The biggest benefit is that it is tractable, as evidenced by the obtained expressions in this paper. The litmus test of its usefulness will be its ability, or otherwise, to explain some observed behavior, even if only qualitatively. Studies in this direction are ongoing. Finally, an extension of our model to account for aspects of quality of service (QoS) should be brought in to enrich the model. We hope to pursue some of these in future works. References [1] D. M. Kreps and J. Sobel, signaling, Chapter 25 in Handbook of Game Theory, vol. 2, R. J. Aumann and S. Hart, Elsevier Science B.V., 1994, pp

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