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Finding red balloons with split contracts

Manuel Cebrian, Lorenzo Coviello, Andrea Vattani, Panagiotis Voulgaris

2012
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Proceedings of the 44th symposium on Theory of Computing - STOC '12
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The present work deals with the problem of information acquisition in a strategic networked environment. To study this problem, Kleinberg and Raghavan (FOCS 2005) introduced the model of query incentive networks, where the root of a binomial branching process wishes to retrieve an information -known by each node independently with probability 1 /n -by investing as little as possible. The authors considered fixed-payment contracts in which every node strategically chooses an amount to offer its
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... mount to offer its children (paid upon information retrieval) to convince them to seek the information in their subtrees. Kleinberg and Raghavan discovered that the investment needed at the root exhibits an unexpected threshold behavior that depends on the branching parameter b. For b > 2, the investment is linear in the expected distance to the closest information (logarithmic in n, the rarity of the information), while, for 1 < b < 2, it becomes exponential in the same distance (i.e., polynomial in n). Arcaute et al. (EC 2007) later observed the same threshold behavior for arbitrary Galton-Watson branching processes. The DARPA Network Challenge -retrieving the locations of ten balloons placed at undisclosed positions in the US -has recently brought practical attention to the problems of social mobilization and information acquisition in a networked environment. The MIT Media Laboratory team won the challenge by acting as the root of a query incentive network that unfolded all over the world. However, rather than adopting a fixed-payment strategy, the team implemented a different incentive scheme based on 1 /2-split contracts. Under such incentive scheme, a node u who does not possess the information can recruit a friend v through a contract stipulating that if the information is found in the subtree rooted at v, then v has to give half of her own reward back to u. Motivated by its empirical success, we present a comprehensive theoretical study of this scheme in the game theoretical setting of query incentive networks. Our main result is that split contracts are robust -as opposed to fixed-payment contracts-to nodes' selfishness. Surprisingly, when nodes determine the splits to offer their children based on the contracts received from their recruiters, the threshold behavior observed in the previous work vanishes, and an investment linear in the expected distance to the closest information is sufficient to retrieve the information in any arbitrary Galton-Watson process with b > 1. Finally, while previous analyses considered the parameters of the branching process as constants, we are able to characterize the rate of the investment in terms of the branching process and the desired probability of success. This allows us to show improvements even in other special cases. case in the Nash equilibrium as our results show. The details of the original model introduced in [12] follow. QUERY INCENTIVE NETWORKS. The scenario of interest is that of a node, the root, that is willing to invest some amount r * to retrieve certain information from a large network in which every node plays strategically. The main goal is to characterize the tradeoff between the investment and the rarity of the information. The model, introduced by Kleinberg and Raghavan [12] , is as follows: the querier node is the root of an infinite d-ary tree, where each node possesses independently the desired information with probability 1 /n, where n represents the rarity of the answer. The root offers each child u a "fixed-payment" contract of r * , stipulating that the root will pay u that amount upon u providing the answer. The query propagates down the tree according to the following scheme: every node u has an integer-valued function f u encoding its strategies; if u is offered a reward of r by its parent and does not possess the answer, then in turn it offers a reward of 1 ≤ f u (r) ≤ r − 1 to its children. When the answer to the query is found, the root selects for payment one among the answer-holders using a fixed non-strategic rule. The payment is then propagated down through the path to that selected node, with each node along the path pocketing its share. If an intermediate node u on this path was offered r by its parent, then its overall payoff is r − f u (r) − 1, where the unit cost is associated with the act of returning the answer 3 . The game-theoretical aspect of the model is that any node u chooses the function f u so to maximize its payoff. To break ties, it is assumed that a node who is offered a reward of one (and does not possess the answer) will always forward the query to its children, even if its expected payoff is zero (since the unit reward would be spent when returning the answer up to its parent). As pointed out in [12] , there is a subtle deficiency with a deterministic tree: the Nash equilibria of a game played in a deterministic network tacitly assume that the nodes know the entire network. Indeed, in a Nash equilibrium, each node chooses its best strategy by knowing the strategies of every other node. However, this is unrealistic, as we want to model a setting where nodes are only aware of their neighbors. To deal with this technical issue, Kleinberg and Raghavan consider a network that can be thought as a branching process from the root. In particular, the number of children of each node is chosen independently from a binomial distribution Bin(d, q), where q is a constant probability of a node being present. The expected number of children of a node -i.e., the branching factor -is then b = qd. By classical results in the theory of branching processes, if b < 1 the process dies out almost surely; therefore there is no amount that the root can offer to obtain an answer with constant probability if the rarity n of the answer is large enough. Instead, for any b > 1, there is a constant non-zero probability that the process will generate infinitely many nodes, so that the answer is present within the first O(log n) levels of the tree with high probability. Nevertheless, Kleinberg and Raghavan show that in the Nash equilibrium the investment needed at the root can be much larger than logarithmic in n. Specifically, while an investment r * = O(log n) is sufficient to retrieve the answer with constant probability for b > 2, an investment of r * = n Θ(1) is needed when 1 < b < 2. That is, in the latter case the root must invest a reward that is exponentially larger than the expected distance from the closest answer. Arcaute et al. [1] generalized the work in [12] showing that this threshold behavior at b = 2 still holds for arbitrary Galton-Watson branching process. They also proved that in a ray -a deterministic infinite path (b = 1, but with zero extinction probability) -the reward needed is super-exponential in the expected depth of the search tree, that is r * = Ω(n!). Finally, they observed that this threshold behavior vanishes if the root desires to find the answer with probability tending to 1: if the desired probability is 1 − 1/n, then

doi:10.1145/2213977.2214047
dblp:conf/stoc/CebrianCVV12
fatcat:nwy5dsfi7vfjjc53ngvguhhxsi