Mechanisms with Referrals: VCG Mechanisms and Multi-Level Mechanisms

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1 Mechanisms with Referrals: VCG Mechanisms and Multi-Level Mechanisms Joosung Lee February 2016 Abstract We study mechanisms for asymmetric information on networks with the following features: 1) A mechanism designer is directly connected to only a part of the whole agents; 2) The agents who are not directly connected to the designer can participate only through referrals; 3) Feasible allocations are depend on the participants induced by others referrals; and 4) In addition to a payoff type, each agent has another private information, a connection type, which represents the agent s acquaintances who can be referred. Thus agents can strategically conceal their network connections by avoiding making referrals. First, we generalize VCG mechanisms. A VCG mechanism is incentive compatible and individually rational if the social welfare is monotone in participants, but it generically runs deficit as it requires too much compensation for referrals. Alternatively as a budget surplus mechanism, we introduce a multi-level mechanism, in which each agent is compensated by the agents who would not be able to participate without his referrals. Under a multilevel mechanism, we show that fully referring one s acquaintances is a dominant strategy and agents have no incentive to under-report their payoff types if the social welfare is monotone and submodular. As applications, a single-item auction and a public good provision game are studied. keywords mechanism design, referral program, reward scheme, VCG mechanisms, multilevel mechanisms, incentive compatibility, budget balancedness. JEL classification: D82, D71, C72 1 Introduction We study mechanisms for environments in which network connections are private information. Such environments with asymmetric information on networks have the following features: i) A mechanism designer, or a planner, is directly connected to only a part of the whole agents. Business School, University of Edinburgh, 29 Buccluech Place, Edinburgh EH8 9JS, United Kingdom. joosung.lee@ed.ac.uk. This paper is based on the third chapter of my Ph.D. dissertation submitted to the Pennsylvania State University. I am grateful to Kalyan Chatterjee for his guidance, encouragement, and support. I also thank Ed Green, Jim Jordan, Vijay Krishna, and Neil Wallace for their helpful comments. All remaining errors are mine. 1

2 ii) The agents who are not directly connected to the designer can participate in a mechanism only through a connected chain of others referrals. iii) Feasible allocations that the designer can choose are depend on the actual participants induced by referrals. iv) Each agent has two-folded private information; a payoff type which represents the agent s preference over the allocations, and a connection type which captures the agent s acquaintances who can be referred. While taking payoff types as private information is standard in the mechanism design literature, the novel part of this paper is considering connection types as another dimension of private information. By private information on connection types, we mean that the designer and the agents know about who are directly connected to themselves, but do not know who are connected to others. Thus agents can strategically conceal their network connections by avoiding making referrals. As a benchmark for an efficient mechanism, first, we generalize a VCG mechanism, which was introduced by Vickrey (1961), Clarke (1971), and Groves (1973). A VCG mechanism is efficient, but neither necessarily incentive compatible nor individually rational. We show that a VCG mechanism is incentive compatible and individually rational if a social welfare increases as the participants increases. This condition has been ignored in the literature, as the population is assumed to be fixed and this population monotonicity is nothing but trivial. It is not surprising that agents may not fully reveal their connection type or may not want to participate in the mechanism without the population monotonicity. Under the monotonicity condition, however, agents truthfully report their payoff types and fully refers their acquaintances. We also show that if a VCG mechanism is incentive compatible then the population monotonicity must hold. While a VCG mechanism is incentive compatible and individually rational under the mild condition; it turns out a VGC mechanism generically runs deficit where the participants are endogenously determined by referrals. We characterize the necessary and sufficient condition for a VCG mechanism to yield a negative surplus to the mechanism designer. Roughly speaking, if agents marginal contribution to the society by referring others is greater than a certain level, then the mechanism designer has to provide too much compensation to the agents who contribute to other agents participation and the compensation dominates the agents total payments. This condition is very mild so that if the core of the network-restricted per-capita game 1 is nonempty, then a VCG mechanism runs deficit. 1 We define this in Section 3.2 based on Myerson (1977). 2

3 In order to overcome the budget problem of a VCG mechanism, we propose an alternative mechanism, namely a multilevel mechanism. 2 Under this mechanism, each agent s payment is based not only on his own marginal contribution but also on that of other agents who are referred to the mechanism designer through him; and he receives some compensation from the referred agents. Under a multilevel mechanism, we show that fully referring one s acquaintances is a dominant strategy and agents have no incentive to under-report their payoff types if the social welfare function is monotone and submodular. In addition, we characterize agents beliefs in which they truthfully report their type, so that for all the agents to tell the truth is a maximin strategy. We provide two applications: a single-item auction and a public good provision game. In standard auctions, for any agent, revealing his connection type is a strictly dominated strategy, since revealing increases competition and there is no compensation on it. A VCG mechanism is efficient but it runs deficit. We show that a multilevel mechanism guarantees a revenue higher than any standard auction. In a public good provision game, a principal can achieve an efficient and budget-balanced allocation by strategically approaching only one agent and letting the agent refer the other remaining agents if agents have a binary preference. While referral programs are prevalent phenomena in various markets and they have been widely studied in the literature of marketing, mechanism design in networks, due to recent development of internet technology and social media, becomes a popular research area in economics and computer science as well. Since Megiddo (1978), many papers, including Ågotnes et al. (2009) and Bachrach and Rosenschein (2007), adopt cooperative game solution concepts to allocation problems in networks and analyze incentive problems. In particular, Hougaard and Tvede (2012) and Hougaard and Tvede (2015) study incentive compatibility on allocation mechanisms in minimum cost spanning networks. In those models, as a network is fixed and its structure is commonly known, referring other agents is not a strategic consideration. Another strand of literature studies incentives on referrals. Lee and Driessen (2012) study agents referral incentives in a variable population environment, proposing a new cooperative solution, namely sequentially two-leveled egalitarianism, and comparing it to Shapley value. Recently computer science literature, such as Yu and Singh (2003), Singh et al. (2011), Emek et al. (2011), and Drucker and Fleischer (2012), deals with multi-level mechanisms in which small rewards are allocated to mitigate free-riding problems or Sybil attacks in peer-to-peer systems. 2 Note that no mechanism is incentive compatible, individually rational, and budget surplus at the same time due to Green and Laffont (1977). 3

4 The paper is organized as follows. Section 2 describes a model with asymmetric information on networks in which agents strategic refer others. In Section 3, we show that a VCG mechanism satisfies both incentive compatibility and individual rationality under a mild condition, but it runs deficit in general. In Section 4, we propose a multilevel mechanism, which guarantees a budget surplus, and characterize conditions for incentive compatibility and individual rationality. In Section 5, we apply our main results to a single-item auction and a public good provision game. Section 6 concludes this paper with a remark related to future research. 2 A model Let N = {1, 2,, n} be a set of (potential) agents. Each agent i N has two-folded private information: a payoff type t i in a payoff type space T i R and a connection type e i in a connection type space E i 2 N\i. Let 0 be a mechanism designer, whose connection type is e 0 E 0 2 N\i. We denote that N 0 = N {0}, T = i N T i, and E = i N0 E i. When there is no danger of confusion, we denote t = (t i, t i) whereas t = (t i, t i ). A typical element e E is a connection profile. Let Γ be a set of allocations. For any S N, Γ(S) is a set of allocations restricted on S. Given t i T i, an agent i s preference over allocations is represented by an utility function, v i (γ t i ) for all γ Γ. Note that an agent s preference is independent on both others private information and his connection type. We assume that, for all i N, all t i T i, and all γ Γ, v i (γ t i ) is finite. Agents are asked to report their payoff types and connection types. An agent whose private information is (t i, e i ) may report any t i T i and any e i e i. Reporting a connection type e i e i to the mechanism designer can be interpreted as referring the members in e i from the set of acquaintances e i. Given a reported (or referred) connection profile e E, the set of participants N(e ) is determined endogenously, [ ] [ ] N(e ) = e 0 i e 0 e i [ j] i e j e e i { 0 } = i N a finite sequence (j 0 = 0, j 1,, j k = i) s.t.( l = 1,, k)j k e j k 1 A connection profile e is trivial if N(e) = e 0 = N, that is a mechanism designer is directly connected to all the players. A mechanism (g, P ) is a pair of an allocation rule g : T E Γ and a payment rule P : T E R n satisfying the following three conditions: i) g(t, e) Γ(N(e)) for all (t, e) T E, that is, the allocation g(t, e) is feasible given a set of participants N(e); ii) P i (t, e) = 0 for all (t, e) T E and all i N \ N(e), that is, a non-participant pays nothing; and 4

5 iii) t N(e) = t N(e) implies g(t, e) = g(t, e) for all e E, that is, the allocation g(t, e) is determined only by the participants report. Under a mechanism (g, P ), if agents report t T and e E, then the agent i whose payoff type is t i gets the payoff v i (g(t, e ) t i ) P i (t, e ). A mechanism (g, P ) is incentive compatible in dominant strategies, if for all (t, e) T E, all i N(e), all t i T i, and all e i e i, v i (g(t, e) t i ) P i (t, e) v i (g(t i, t i, e i, e i ) t i ) P i (t i, t i, e i, e i ). An agent s utility from not participating is normalized to zero. A mechanism (g, P ) is individually rational, if for all (t, e) T E and all i N(e), v i (g(t, e) t i ) P i (t, e) 0. Given a mechanism (g, P ) and a profile of private information (t, e) T E, the sum of agents payments P i(t, e) is the ex-post surplus to the mechanism designer. A mechanism (g, P ) runs deficit for (t, e) T E, if the ex-post surplus to the mechanism designer is negative. For all t T and all S N, a social welfare function 3 is defined by W (t, S) = max v i (γ t i ). γ Γ(S) Given t T and S N, a set of efficient allocations is { Γ (t, S) γ Γ(S) } v i (γ t i ) = W (t, S). i S i S An allocation rule g is efficient if, for all (t, e) T E, g(t, e) Γ (t, N(e)). We close this section by providing a condition for implementability, extending the result of Rochet (1987), which imposes a version of monotonicity of preferences. With asymmetric information on networks, in addition to the Rochet s original condition, it requires for each agent to have bounded preferences. An allocation g is implementable in dominant strategies if there exists a payment rule P such that the mechanism (g, P ) is incentive compatible in dominant strategies. Given e E, i N(e), and t i T i, define a length l between t i and t i in T i : l(t i, t i ) = v i (g(t, e) t i ) v i (g(t i, t i, e) t i ) t i, t i T i. Proposition 1. An allocation rule g is implementable in dominant strategies if and only if 3 For simplicity, we assume that no external cost is required for each allocation γ Γ. One can define a welfare function including a cost if it is required. 5

6 i) the l-weighted graph of T i has no finite cycle of negative length for all e E, all i N(e), and all t i T i ; and ii) v i (g(t i, t i, e) t i ) is finite for all t T, all e E, all i N(e), and all t i T i. The proof is omitted as it can be easily obtained based on Rochet (1987) or Vohra (2011). 3 A VCG mechanism In this section, we generalize a VCG mechanism in asymmetric information on networks. In complete information on networks, in which the mechanism designer knows all the agents directly so that e 0 = N(e), any referral does not effect on the allocation and the payment at all. In this trivial connection profile case, a VCG mechanism consists of an efficient allocation rule g and a payment rule P such that P i (t, e) = W (t, N(e) \ {i}) v j (g(t, e) t j ), (1) j N(e)\{i} which represents an agent i s marginal effect on the other agents. In fact, it does not depend on e i. It can be easily shown that revealing agents true types is a dominant strategy and the payment rule implements the efficient allocation rule as long as e 0 = N(e). Now consider non-trivial cases. An agent affects the economy in two ways: a change in a allocation due to a payoff type within a fixed participants and a change in participants due to referrals. However, in cases that referrals increases the participants, that is, e 0 N(e), the above payment rule does not guarantee fully referring. Hence, we should take the effect of referrals into account. 3.1 Contribution on New Participants Given e E and i N, the set of i s outsiders, O i (e), consists of agents who can be involved without i, that is O i (e) N(e i ) \ {i}. Note that e is trivial if and only if O i (e) = N(e) \ {i} for all i N. The agent i s group, G i (e), consists of agents who can not participate without i, that is G i (e) N(e) \ O i (e). Note that e is trivial if and only if G i (e) = {i} for all i N. Given e E, i is a predecessor of j if j G i (e). Note that N(e) endowed with this precedence relation is a tree and we call it a contribution tree. For all i N, the set of i s predecessors is R i (e) {j N i G j (e)} and the set of i s referred agents is F i (e) e i \ O i (e). An agent i s height is the number of predecessors, that is, h i (e) R i (e). For convenience, let R 0 (e) and h 0 (e) 0 for all e E. Let N d (e) {j N(e) h j (e) = d} for each 1 d max i N h i (e). Note that {N d (e)} 1 d maxi N h i (e) is a partition of N(e). 6

7 (a) A connection profile e (b) The contribution tree induced by e Figure 1: A connection profile and the induced contribution tree. A connection profile e E represented by a directed graph on N, which possibly contains cycles; however, the induced contribution tree {G i (e)} i N has no cycle. The following example illustrates the relation between a connection profile and the induced contribution tree. Example 1. Let N = {1, 2, 3, 4, 5}. Let e = {e i } i N0 with e 0 = {1, 2}, e 1 = {3, 4}, e 2 = {4}, e 3 = e 4 =, and e 5 = {2}. We have that N(e) = {1, 2, 3, 4} and the induced contribution tree consists of G 0 (e) = {0, 1, 2, 3, 4}, G 1 (e) = {1, 3}, G 2 (e) = {2}, G 3 (e) = {3},G 4 (e) = {4}, and G 5 (e) =. That is, the agent 5 cannot be involved in this connection profile. Figure 1 represents the connection profile e and the induced contribution tree. Note that 4 e 0 but 4 N 1 (e). 3.2 Characterizations of a VCG mechanism We are ready to define a VCG mechanism in asymmetric information on networks. Definition 1 (VCG Mechanism). A VCG mechanism consists of an efficient allocation rule g and a payment rule P V such that P V i (t, e) = W (t, O i (e)) j N(e)\{i} The VCG payment can be decomposed in two parts: Pi V (t, e) = [W (t, N(e) \ {i}) j N(e)\{i} v j (g(t, e) t j ). (2) ] [ ] v j (g(t, e) t j ) W (t, N(e) \ {i}) W (t, O i (e)). The first part is the usual VCG payment in the fixed participants N(e), which is exactly the same with (1); and the second part is a reward for effective referrals. If a connection profile e is trivial, then the second part is always zero to any player. 7

8 In a fixed population setting, it is well-known that a VCG mechanism satisfies both incentive compatibility and individual rationality. When the population is determined by the players referrals, however, a VCG mechanism is neither necessarily incentive compatible nor individually rational when we allow agents to have asymmetric information on networks. We show that a VCG mechanism is incentive compatible and individually rational in dominant strategies if a welfare function W increases as the population increases. Definition 2. A welfare function W is monotone if for all t T, S S = W (t, S ) W (t, S). Proposition 2. If W is monotone, then a VCG Mechanism is incentive compatible and individually rational. Furthermore, if a VCG Mechanism is incentive compatible, then W is monotone. Proof. In Step 1, we show that the monotonicity W is a necessary and sufficient condition for a VCG mechanism to be incentive compatible. In Step 2, we show that the monotonicity W is a sufficient condition for a VCG mechanism to be individually rational. Step 1 : First, suppose that W is monotone. Given t i and e i, consider an agent i whose private information is t i T i and e i E i. Reporting t i T i and e i e i, the agent i s payoff is v i (g(t, e ) t i ) Pi V (t, e ) = v i (g(t, e ) t i ) W (t, O i (e )) v j (g(t, e ) t j ) = v i (g(t, e ) t i ) + j N(e )\{i} j N(e )\{i} v j (g(t, e ) t j ) W (t, O i (e)) W (t, N(e )) W (t, O i (e)) W (t, N(e)) W (t, O i (e)). The second equality is from independence of private information, that is, W (t, O i (e )) = W (t, O i (e)). The third line is from the fact that v i (g(t, e ) t i )+ j N(e )\{i} v j(g(t, e ) t j ) is maximized at t i = t i. The last inequality is from the monotonicity of W and the condition e i e i. Thus reporting t i and e i is a dominant strategy for the agent i. Conversely, suppose that the VCG mechanism is incentive compatible but W is not monotone. There exist t T and S, S N such that S S = and W (t, S) > W (t, S S ). Since either S or S must be nonempty, let S and choose k S. Consider a connection profile e such that e 0 = S, e k = S, and e i = for all i 0, k. Let e k =. Since N(e) = S S, N(e ) = S, and O k (e) = O k (e ), incentive compatibility implies that W (t, S S ) W (t, S), 8

9 which contradicts to monotonicity. Step 2 : Suppose that W is monotone. Since reporting one s private information truthfully is a dominant strategy from Step 1, it suffices to show, for all (t, e) T E and all i N, v i (g(t, e) t i ) P V i (t, e) 0 for individual rationality. Definition of W yields that v i (g(t, e) t i ) P V i (t, e) = W (t, N(e)) W (t, O i (e)). However, by monotonicity of W, W (t, N(e)) W (t, O i (e)) is nonnegative. Next, we observe a mechanism designer s budget problem. It turns out a VCG mechanism tends to run deficit under a mild condition. The following proposition shows that if the welfare of the economy is greater than the sum of each agent s per capita welfare then a VCG mechanism runs deficit. Proposition 3. Given (t, e) T E, a VCG mechanism (g, P V ) yields nonnegative surplus if and only if Proof. We have that P V i (t, e) = = + = W (t, N(e)) W (t, O i (e)) = W (t, O i (e)) v i (g(t, e) t i ) W (t, O i (e)) N(e) 1. j N(e)\{i} j N(e)\{i} v j (g(t, e) t j ) v j (g(t, e) t j ) v i (g(t, e) t i ) [W (t, O i (e)) W (t, N(e))] + W (t, N(e)) (3) W (t, O i (e)) ( N(e) 1)W (t, N(e)). Thus, we have P i V (t, e) 0 if and only if W (t, N(e)) W (t,o i (e)) N(e) 1, as desired. The following corollary provides a more specific necessary condition for a VCG mechanism to run deficit. Roughly speaking, if an agent substantially increases the social welfare by refering some other agent, then it runs deficit. Corollary 1. Suppose that W is monotone. Given (t, e) T E, if there exist j, k N(e) such that then a VCG mechanism runs deficit. W (t, N(e)) > W (t, O j (e)) + W (t, O k (e)), (4) 9

10 Proof. Plugging the condition (4) into that (3), we have that P V i (t, e) < \{j,k} where the second inequality is due to monotonicity. [W (t, O i (e)) W (t, N(e))] 0, Example 2. A seller allocates an indivisible good via a VCG mechanism to two existing agents, say X and Y, and their valuations to the item are x and y, respectively. Suppose X can refer another potential agent, say Z, whose valuation is z. A VCG mechanism runs deficit if and only if max{x, y, z} > y + max{x, y}. Next corollary investigates the relation between the budget problem of a VCG mechanism and the existence of core of the underlying environment. According to the notion of network-restricted games proposed by Myerson (1977), given (t, e) T E, we define a network-restricted per-capita game (N(e), v t,e ), where W (t,n(e)) N(e) if S = N(e) v t,e W (t,o (S) = T (e)) N(e) T if S = N \ T 0 otherwise. If the connection profile e is trivial, then the network-restricted per-capita game is equivalent to the per-capita game. Corollary 2. Given (t, e) T E, if a VCG mechanism runs strict surplus, then the core of (N(e), v t,e ) is empty. Proof. Suppose that the core of (N(e), v t,e ) is nonempty. Due to Bondareva (1963) and Shapley (1967), for all probability measure δ on 2 N(e) \ {, N(e)}, Now define δ by: S N(e) δ(s) = δ(s)v t,e (S) v t,e (N(e)) = { 1 N(e) 0 otherwise. With this weight δ, the left hand side of (5) becomes S N(e) δ(s)v t,e (S) = W (t, N(e)). (5) N(e) if S = N(e) \ {i} and i N(e) 1 W (t, O i (e)) N(e) N(e) 1. (6) By (5), (6), and Proposition 3, it contradicts that a VCG mechanism runs strict surplus. 10

11 4 A Multilevel mechanism This section introduce an alternative mechanism, a multilevel mechanism, which guarantees a nonnegative surplus to the mechanism designer. As we have seen in the previous section, under a VCG mechanism the mechanism designer provides too much compensation to agents who contribute to new agents participation, and hence it generically runs deficit. Under a multilevel mechanism, an agent is compensated with an appropriate transfer not from the mechanism designer but from his referred agents, who would not be able to participate without him. Definition 3 (Multilevel Mechanism). A multilevel mechanism consists of an efficient allocation rule g and a payment rule P M which satisfies, for all i N(e), where P G i (t, e) = W (t, O i (e)) Pi M (t, e) = Pi G (t, e) j O i (e) j F i (e) v j (g(t, e) t j ). P G j (t, e), Remark. Note that Pi G (t, e) is the marginal effect of the agent i s group on the economy. This is different from the VCG payment Pi V (t, e), which is the marginal effect of i alone. Under a multilevel mechanism, an agent i pays Pi G (t, e) to i s immediate predecessor. Only the agents in N 1 (e) pay to the mechanism designer. That is, each agent is responsible for his group when he pays to his predecessor, but he is compensated with some transfer from his referred agents. Following lemma shows that the mechanism designer s net surplus equals to the sum of the marginal effects of the agents group, whose height is 1. Lemma 1. For all (t, e) T E, we have P M i (t, e) = i N 1 (e) P G i (t, e). (7) Proof. Since Pi M (t, e) = Pi G(t, e) j F i (e) P j G (t, e), we have that Pi M (t, e) = Pi G (t, e) Pj G (t, e). (8) j F i (e) Let D = max i N h i (e). Since {N d (e)} 1 d D is a partition of N(e), the last term of (8) can be written by: Pj G (t, e) = j F i (e) = D Pj G (t, e) d=1 i N d (e) j F i (e) D Pj G (t, e), (9) d=2 i N d (e) 11

12 where the second equality follows from the two facts; {F i (e)} i N d (e) is a partition of N d+1 (e) for all 1 d D 1, and F i (e) = for all i N D (e). Plugging (9) into (8), we have the desired result. The following proposition shows that a multilevel mechanism guarantees a nonnegative revenue. Proposition 4. A multilevel mechanism is budget surplus Proof. Take any (t, e) T E. By Lemma 1, it suffices to show i N 1 (e) P i G (t, e) 0. By the definition of W, for all i N(e), we have that Thus, i N P M i P G i (t, e) = W (t, O i (e)) j O i (e) (t, e) P i M (t, e) 0, as desired. v j (g(t, e) t j ) 0. (10) Our main results related to agents incentives in a multilevel mechanism depend on the structure of underlying environments. We define submodularity of an individual player s preference and an aggregated welfare function. Definition 4. A utility function v i is submodular if t i t i and v i(γ t i ) v i (γ t i ) implies v i (γ t i ) v i (γ t i ) v i (γ t i) v i (γ t i). Definition 5. A welfare function W is submodular (in coalitions) if for all S, S N and all t T W (t, S) + W (t, S ) W (t, S S ) + W (t, S S ). The following lemma characterizes submodularity of a general set function, namely Exclusion-Inclusion Principal for Submodularity. Lemma 2. A set function W defined on 2 N is submodular if and only if, for all S, S N such that S S and all partition P of S \ S, W (S) W (S ) P P [W (S) W (S \ P )]. (11) Proof. First, we prove the only-if part by mathematical induction. Let S be an arbitrary subset of N. If S = S, then the claim is obvious. Suppose that S \ S is singleton, say S \ S = {k}. Then the claim (11) trivially holds. As induction hypothesis, suppose that, for all S S such that S \ S l, the statement (11) is true. Now consider a subset S S such that S \ S = l + 1 and a partition P of S \ S. Pick any k S \ S and let P 1 P such that k P 1. Define a new partition P of S \ (S {k}) by replacing P 1 with P 1 P 1 \ {k}, that is, P = (P \ {P 1 }) { P 1 }. Note that P 1 is possibly empty. Since S \ (S {k}) l, we have that W (S) W (S {k}) [W (S) W (S \ P )]. P P 12

13 Subtracting W (S ) W (S {k}) from the both side, we have that W (S) W (S ) P P [W (S) W (S \ P )] [W (S ) W (S {k})] = P P\{P 1 } [W (S) W (S \ P )] +W (S) W (S \ P 1 ) [W (S ) W (S {k})]. (12) On the other hand, by submodularity, we have W (S {k}) + W (S \ P 1 ) W (S ) + W (S \ P 1 ). (13) Plugging (13) into (12), we have W (S) W (S ) [W (S) W (S \ P )], P P which implies that the claim (11) holds for S S such that S \ S = l + 1 and the given P. Since we have chosen S and P arbitrarily, we get the desired conclusion. Next, we prove the if part. Pick any U, V N. Let S = U V, S = U V, and P = {U \ S, V \ S }. The condition (11) implies that W (S) W (S ) [W (S) W (V )] + [W (S) W (U)], or equivalently, W (U) + W (V ) W (S) + W (S ), as desired. For notational simplicity, given (t i, e i ) T i E i and t i T i, define functions w i : T i E i R and u i : T i E i R as follows: w i (t i, e i) = [ W (t, N(e )) W (t, O i (e )) ] u i (t i, e i t i ) = v i (g(t, e ) t i ) v i (g(t, e ) t i). j F i (e ) [ W (t, N(e )) W (t, O j (e )) ] Note that w i (t i, e i ) represents the difference between the marginal contribution of i s referrals and the sum of the marginal contributions of i s referred agents referrals. On the other hand, u i (t i, e i t i) represents i s additional gain of misreporting t i instead of t i from what he would get if he were a type of t i. In the following Lemma, we will decompose an agent s ex-post payoff into the sum of w i (t i, e i ) and u i(t i, e i t i). Lemma 3. Suppose that an agent i s type is (t i, e i ) T i E i and others report (t i, e i ) T i E i. Under the multilevel mechanism, i) the agent i s ex-post payment from truth telling is P M i (t, e) = W (t, O i (e)) + j F i (e) [W (t, N(e)) W (t, O j (e))] j N(e)\{i} v j (g(t, e) t j )); 13

14 ii) the agent i s ex-post payoff from truth telling is, v i (g(t, e) t i ) P M i (t, e) = w i (t i, e i ); iii) the agent i s ex-post payoff from reporting (t i, e i ) T i E i is v i (g(t, e ) t i ) P M i (t, e ) = w i (t i, e i) + u i (t i, e i t i ); Proof. Note that P M i (t, e) = P G i (t, e) = = j F i (e) W (t, O i (e)) W (t, O i (e)) j F i (e) = j N(e)\{i} P G j (t, e) j O i (e) j O i (e) v j (g(t, e) t j ) v j (g(t, e) t j ) j F i (e) [W (t, O j (e)) W (t, N(e))] v j (g(t, e) t j )) + W (t, O i (e)) + j F i (e) k G j (e) j F i (e) W (t, O j (e)) v k (g(t, e) t k ) k O j (e) [W (t, N(e)) W (t, O j (e))], v k (g(t, e) t k ) where the third equality is due to k O j (e) v k(g(t, e) t k ) = W (t, N(e)) k G j (e) v k(g(t, e) t k ) and the last equality is from O i (e) ( j Fi (e)g j (e) ) = N(e) \ {i}. This completes the first part. Using the first part, we have v i (g(t, e) t i ) P M i (t, e) = W (t, N(e)) W (t, O i (e)) = w i (t i, e i ), which completes the second part. j F i (e) [W (t, N(e)) W (t, O j (e))] When the agent i reports (t i, e i ) T i E i, his payoff is v i (g(t, e ) t i ) Pi M (t, e ). Subtracting and adding v i (g(t, e ) t i ), we have v i (g(t, e ) t i ) P M i (t, e ) = v i (g(t, e ) t i ) v i (g(t, e ) t i) + v i (g(t, e ) t i) P M i (t, e ) which completes the last part. = u i (t i, e i t i ) + w i (t i, e i), Proposition 5. A multilevel mechanism is individually rational if W is monotone and submodular. 14

15 Proof. Due to Lemma 3, it suffices to show that w i (t i, e i ) 0 if W is monotone and submodular. Suppose that W is monotone and submodular. Take any (t, e) T E. Since {{G j (e)} j Fi (e), {i}} is a partition of N(e) \ O i (e), applying Lemma 2, we have that W (t, N(e)) W (t, O i (e)) [W (t, N(e)) W (t, N(e) \ G j (e))] + [W (t, N(e)) W (t, N(e) \ {i})]. (14) j F i (e) Since W is monotone, W (t, N(e)) W (t, N(e) \ {i}) 0, the inequality (14) yields that [W (t, N(e)) W (t, O i (e))] j F i (e) which is equivalent that w i (t i, e i ) 0. [W (t, N(e)) W (t, N(e) \ G j (e))], Next, we investigate conditions for incentive compatibility with respect to a connection type and a payoff type. Under a multilevel mechanism, first we show that agents have no incentive to under-report their payoff types. Given this result, it turns out that fully referring is a dominant strategy. We define a submodularity in payoff-types of a welfare function. Definition 6. A welfare function W is submodular in payoff-types if, for all t, t T and all S N, W (t, S) + W (t, S) W (t t, S) + W (t t, S), where t t = (max{t i, t i }) i N and t t = (min{t i, t i }) i N. Proposition 6. For all (t, e) T E and all i N(e), under-reporting i s payoff type is dominated by reporting the true type t i if W is submodular in payoff-types. Proof. Given (t, e) T E and i N(e), due to Lemma 3, the agent i s payoff from reporting t i T i, v i (g(t, e) t i ) Pi M (t, e), consists of three parts: [ W (t, N(e)) W (t, O j (e)) ] v i (g(t, e) t i ) + j N(e)\{i} } {{ } (A) v j (g(t, e) t j )) W (t, O i (e)) }{{} (B) j F i (e) } {{ } (C) (15) The first part (A) is maximized at t i = t i and the second part (B) does not changed by t i. Since W is submodular in payoff-types, for all j F i(e), if t i < t i, then W (t, N(e)) W (t, O j (e)) W (t, N(e)) W (t, O j (e)). Thus, if t i < t i, then the last part (C) is less than or equals to which is from truth-telling. j F i (e) [W (t, N(e)) W (t, O j (e))],. 15

16 We define a competitiveness of an underlying environment. Definition 7. An environment (v, Γ) is competitive if, for all S, S N, all t T, all γ Γ (t, S), and all γ Γ (t, S ), S S = v i (γ t i ) v i (γ t i ) i S, t i T i. Proposition 7. Suppose that W is submodular and (v, Γ) is competitive. Under a multilevel mechanism, it is a dominant strategy for an agent i to fully reveal his connection type, if v i is submodular. That is, for all t i T i and all e i e i, w i (t i, e i ) + u i (t i, e i t i ) w i (t i, e i) + u i (t i, e i t i ), (16) Proof. The proof consists of two steps. In Step 1, we will show that submodularity of W implies that for all t i T i and all e i e i, w i (t i, e i ) w i (t i, e i). (17) In Step 2, we will show that competitiveness of Γ and submodularity of v i imply that for all t i T i and all e i e i, u i (t i, e i t i ) u i (t i, e i t i ). (18) Two conditions, (17) and (18), implies (16), which completes the proof. Step 1 : Rearranging terms, the inequality (17) is equivalent to W (t, N(e)) W (t, N(e ))+ [ W (t, N(e )) + W (t, O j (e)) W (t, N(e)) W (t, O j (e )) ] j F i (e ) [ W (t, N(e)) W (t, O j (e)) ]. (19) j F i (e)\f i (e ) Since N(e ) O j (e) = N(e) and N(e ) O j (e) = O j (e ) for all j F i (e ), submodularity of W implies that [ W (t, N(e )) + W (t, O j (e)) W (t, N(e)) W (t, O j (e )) ] 0. (20) j F i (e ) Applying Lemma 2, submodularity of W implies that W (t, N(e)) W (t, N(e [ )) W (t, N(e)) W (t, O j (e)) ]. (21) j F i (e)\f i (e ) Since (20) and (21) implies (19), we have shown (17). Step 2 : By competitiveness of Γ, we have that v i (g(t, e ) t i ) v i (g(t, e) t i ). Due to Proposition 6, the agnet i reports t i t i in equilibrium, submodularity of v i yields that v i (g(t, e ) t i) v i (g(t, e) t i) v i (g(t, e ) t i ) v i (g(t, e) t i ), which implies (18). 16

17 We provide agents beliefs in which they truthfully report their types, so that for all the agents to tell the truth is a maximin strategy. Proposition 8. Suppose W is monotone. For any player i N, truth-telling of a payoff type is a best response if the player i believes that, for all t i T i, W (t, N(e)) = W (t, O i (e) {i}). (22) Proof. It suffices to show the last part (C) of (15) is zero. Since O i (e) {i} O j (e) N(e) for all j F i (e), monotonicity of W implies that, for all t i T i, W (t, O i (e) {i}) W (t, O j (e)) W (t, N(e)). However, the condition (22) yields that, for all j F i (e), W (t, O j (e)) = W (t, N(e)), as desired. Remark. The condition (22) is induced by two possibilities. First, the agent i has no contribution on new agents participations, that is, F i (e) =. Second, the agent i s group does not increase the social welfare. Therefore, if agents are ambiguity-averse and they cannot exclude such possibilities, then truthfully reporting their payoff types is a dominant strategy. 5 Applications In this section, we provide two interesting applications; a single-item selling problem and a public good provision game. 5.1 Single-item Auctions The seller auctions off an indivisible item and each agent i has a private valuation t i R + to the item. We assume that resale is prohibited. 4 For tractability, we assume that t i t j for all i, j N and i j. Let i (t, S) argmax i S t i. In a second price auction, for all i e 0, it is a dominant strategy to bid his true value and to fully refer his acquaintances. That is, given (t, e) T E, i (t, e 0 ) wins the auction and the seller s ex-post revenue is the second highest value among e 0, REV SP A = max t i. i e 0 \{i (t,e 0 )} If the actual connection profile is not trivial, then a VCG mechanism may be different from a second price auction. Fix (t, e) T E. If i (t, N(e)) e 0, then the seller s 4 This assumption is crucial for agents payoff types and connection types to be independent. 17

18 ex-post revenue from a VCG mechanism equals to the one from a second price auction. If i (t, N(e)) N 1 (e) \ e 0, then a VCG mechanism yields a higher revenue than a second price auction. If i (t, N(e)) N 1 (e), then the seller s revenue from a VCG mechanism is at most equals to the one of a second price auction. Example 3. Let N = {1, 2, 3} and t 1 = 1, t 2 = 2, t 3 = 4, and e 0 = {1, 2}. No matter what {e i } i=1,2,3 is, the ex-post revenue from a second price auction is Suppose e 1 = {3} and e 2 = e 3 =. Since all the agents are involved, agent 3 wins the auction and pays 2. However, the seller should pay 2 to agent 1, because agent 1 s marginal contribution to the economy is 2. Hence, the seller s ex-post revenue is 0, which is less than the case of a second price auction. 2. Suppose e 2 = {3} and e 1 = e 3 =. Again agent 3 wins and pays 2. However, the seller should pay 3 to agent 2, and the seller s ex-post revenue is -1, which runs deficit. 3. Suppose e 1 = e 2 = {3} and e 3 =. Since agent 3 is referred by both agent 1 and agent 2, none of them can claim an exclusive contribution on agent 3 s participation and 3 N 1 (e). The seller s ex-post revenue is 2, which is higher than the case of a second price auction. While the revenue ranking between a second price auction and a VCG mechanism depends on a connection profile, a multilevel mechanism guarantees a revenue higher than or equal to a second price auction. Lemma 4. A multilevel mechanism guarantees a revenue higher than or equal to a second price auction for single-item cases. Proof. Fix (t, e) T E. Since all agents fully reveal their connection type, N(e) is the set of participants. Let ˆt be an equilibrium bidding strategy profile. Then i (ˆt, N(e)) is the winner. There exists i 1 N 1 (e) such that i (ˆt, N(e)) G i 1(e), that is, i 1 is the existing buyer who contributes the winner s participation. Using Lemma 1, the seller s net revenue is REV M = Since N 1 (e) \ {i 1 } O i 1(e) and ˆt i (ˆt,N 1 (e)) t i 1, max ˆt i i O i 1 (e) Since e 0 N 1 (e), we have max ˆt i i N 1 (e)\{i 1 } max ˆt i. i O i 1 (e) max ˆt i. i N 1 (e)\{i (ˆt,N 1 (e))} max ˆt i max ˆt i. i N 1 (e)\{i (ˆt,N 1 (e))} i e 0 \{i (ˆt,e 0 )} 18

19 The remaining proof is divided into two cases: Case 1: i (t, e 0 ) = i (ˆt, e 0 ). By Proposition 6, since ˆt i t i for all i N(e), we have which equals to REV SP A. Case 2: i (t, e 0 ) i (ˆt, e 0 ). Then, we know By Proposition 6 again, we have which is greater than REV SP A. max ˆt i max t i, i e 0 \{i (ˆt,e 0 )} i e 0 \{i (t,e 0 )} max ˆt i ˆt i (t,e 0 ). i e 0 \{i (ˆt,e 0 )} ˆt i (t,e 0 ) t i (t,e 0 ), However, this result is not necessarily true for combinatorial auctions. The following example shows that a multilevel mechanism may yield a lower revenue than a second price auction. Example 4. The seller auctions two item, X and Y. She knows buyer 1 and buyer 2 directly, but not buyer 3. Buyer 3 is only known to buyer 2. That is e 0 = {1, 2}, e 1 = {3}, and e 2 = e 3 =. Their values to the items are as follows: Buyer X Y X + Y Second price auction: Buyer 1 does not refer buyer 3. Buyer 1 and buyer 2 bid their true values. Buyer 1 wins Y and pays nothing; buyer 2 wins X and pays 6. Hence, the seller s revenue is VCG mechanism: Buyer 1 refers buyer 3. All bidders bid their true values. Buyer 2 wins X and pays 4; buyer 3 wins Y and pays 4. Buyer 1 gets a transfer 8 from the seller, and the seller s revenue is Multilevel mechanism: Buyer 1 refers buyer 3. Suppose that all bidders bid their true values. Buyer 2 wins X and pays 4 to the seller; buyer 3 wins Y and pays 4 to buyer 1. Hence, the seller s revenue is 4, which is less that the case of a second price auction. Now, we compare a multilevel mechanism and a standard auction in terms of the seller s expected revenue. An auction is standard if the highest bidder wins the item, the expected payment of a bidder with value zero is zero, and there is no compensation on referrals. The following proposition shows that the expected revenue from a VCG mechanism exceeds that from any standard auction. 19

20 Proposition 9. Suppose that values are independently and identically distributed and all bidders are risk neutral. A multilevel mechanism dominates standard auctions in terms of seller s ex-post revenue for single-item cases. Proof. By Lemma 4, for all (t, e) T E, a multilevel mechanism guarantees a revenue higher than or equal to a second price auction. Thus, for any seller s belief on (t, e), the seller s expected payoff from a multilevel mechanism is higher than or equals to the one of a second price auction. Applying the revenue equivalence principle 5, the seller s expected payoff from a multilevel mechanism is higher than or equals to the one from any standard auction with the set of bidders e Public Good Provision Since a public good is both non-excludable and non-rivalrous, it is the ideal example with which agents connection types and payoff types are independent. It is known that there is no budget-balanced mechanism which satisfies both incentive compatibility and individual rationality. However, allowing strategic referrals, a multilevel mechanism turns out to balance the budget. Especially, if each agent has a binary preference, then it satisfies both incentive compatibility and individual rationality. Example 5 (Public good provision). A principal decides whether or no to provide a public good at a fixed cost c (1, 2). There are two agents whose private value is either 0 or 1. That is, providing the public good is efficient if and only if both agents values are 1. Suppose that both the principal and agents can directly contact with others and that both agents reveal their true types if they are indifferent between truth-telling and lying. 1. A VCG mechanism with both agents: For both agents, revealing their true values is a dominant strategy and the allocation is efficient. When both agents values are 1, each pays just c 1 and hence the budget is 2(c 1) c = c 2 < A VCG mechanism with a randomly chosen agent: For the chosen agent, contacting with the other agent is a dominant strategy if her value is 1. Again, for both players, revealing their true value is a dominant strategy. The outcome is exactly the same with the previous case. 3. A multilevel mechanism with a randomly chosen agent: For the chosen agent, contacting with the other agent is a dominant strategy if his value is 1. For both players, revealing their true value is a dominant strategy. Suppose that both agents values are 1. For the chosen agent, she pays c to the principal and compensated with c 1 from the other agent. Thus the allocation is efficient and balances the budget. 5 Myerson (1981) shows that any symmetric and increasing equilibrium of any standard auction yields the same expected revenue to the seller. 20

21 6 Concluding Remarks In this paper, we considered agents asymmetric knowledge of information on others connections in mechanism design problems. Allowing asymmetric information on networks, each agent has an incentive not only to misreport his payoff type but also to conceal his connection type, and hence the participants are endogenously determined. In such generalized environments, a VCG mechanism has good properties in terms of agents incentive problems, but it runs deficit. Alternatively, a multilevel mechanism has been proposed and it guarantees nonnegative surplus. The multilevel mechanism proposed in this paper is of course not the only one which yields nonnegative surplus to the mechanism designer. We can define a class of mechanisms by generalizing the compensation rule. A multilevel mechanism admits an agent s contribution to increasing population only if it is exclusive; that is, if a new agent has been referred by two or more existing agents, then none of them has a right to claim the contribution of the new agent s participation. Many other alternative rules are possible. For instance, all agents who are related to the newcomer can split the contribution according to a certain portion. These generalized mechanisms give agents more incentive to refer other potential agents and still guarantee nonnegative payoffs. References Ågotnes, T., W. van der Hoek, M. Tennenholtz, and M. Wooldridge (2009): Power in normative systems, in Proceedings of The 8th International Conference on Autonomous Agents and Multiagent Systems-Volume 1, International Foundation for Autonomous Agents and Multiagent Systems, Bachrach, Y. and J. S. Rosenschein (2007): Computing the Banzhaf power index in network flow games, in Proceedings of the 6th international joint conference on Autonomous agents and multiagent systems, ACM, 254. Bondareva, O. (1963): Some applications of linear programming methods to the theory of cooperative games, Problemy kibernetiki, 10, Clarke, E. (1971): Multipart pricing of public goods, Public choice, 11, Drucker, F. A. and L. K. Fleischer (2012): Simpler sybil-proof mechanisms for multi-level marketing, in Proceedings of the 13th ACM conference on Electronic commerce, ACM, Emek, Y., R. Karidi, M. Tennenholtz, and A. Zohar (2011): Mechanisms for multi-level marketing, in Proceedings of the 12th ACM conference on Electronic commerce, ACM,

22 Green, J. and J.-J. Laffont (1977): Characterization of satisfactory mechanisms for the revelation of preferences for public goods, Econometrica: Journal of the Econometric Society, Groves, T. (1973): Incentives in teams, Econometrica: Journal of the Econometric Society, Hougaard, J. L. and M. Tvede (2012): Truth-telling and Nash equilibria in minimum cost spanning tree models, European Journal of Operational Research, 222, (2015): Minimum cost connection networks: Truth-telling and implementation, Journal of Economic Theory, 157, Lee, J. and T. Driessen (2012): Sequentially two-leveled egalitarianism for TU games: Characterization and application, European Journal of Operational Research. Megiddo, N. (1978): Computational complexity of the game theory approach to cost allocation for a tree, Mathematics of Operations Research, 3, Myerson, R. (1981): Optimal Auction Design, Mathematics of Operations Research, 6, Myerson, R. B. (1977): Graphs and cooperation in games, Mathematics of operations research, 2, Rochet, J. (1987): A necessary and sufficient condition for rationalizability in a quasilinear context, Journal of Mathematical Economics, 16, Shapley, L. (1967): On balanced sets and cores, Naval research logistics quarterly, 14, Singh, V. K., R. Jain, and M. Kankanhalli (2011): Mechanism design for incentivizing social media contributions, in Social media modeling and computing, Springer, Vickrey, W. (1961): Counterspeculation, auctions, and competitive sealed tenders, The Journal of finance, 16, Vohra, R. (2011): Mechanism Design: a linear programming approach, vol. 47, Cambridge Univ Pr. Yu, B. and M. P. Singh (2003): Incentive mechanisms for peer-to-peer systems, in Agents and Peer-to-Peer Computing, Springer,