1. Query Incentive Networks Jon Kleinberg and Prabhakar Raghavan - Presented by: Nishith Pathak

2. Motivation Understanding networks of interacting agents as economic systems Users pose queries and offer incentives for answers The queries and incentives are propagated in the network Vetting – Nodes along the path validate the relationship between the end-points Can be formulated as a game played by nodes in the network This game has a Nash Equilibrium

3. Motivation In case of users seeking information without incentives the critical behavior is at branching parameter 1 However, for users seeking information with incentives, the critical behavior is at branching parameter 2 Between parameters 1 and 2, the answer is within vicinity but the incentive required is too high

4. Formulating a Model An infinite d-ary tree structure T is assumed With each step the incentive keeps diminishing The set of strategies for every node is the set of functions which decides the split between pay-off and reward to child nodes Parameters –  q : Probability of a node being active given that its parent is active  b = qd : branching factor (Mean number of offsprings) Based on q, only a subset of T, T’ will be active If b<1 then T’ is almost surely finite If b>1 then T’ is infinite with probability, 1-e q,d >0

5. Formulating a Model How much utility r* is required by the root node v* in order to achieve a probability  of obtaining an answer from the network Utility r* depends on probability (1-p) that a node has the answer  1 out of every n nodes have the answer (rarity n of the answer), where n = (1-p) -1 Value on effort  Utilities are dealt as integers only to prevent degenerate case  Every node on the path to the answer has to accept a minimum reward of 1 utility  This is incorporated in the model by placing a value on the communication effort of the node  This minimum utility of 1 does not count towards the payoff Three step process –  Query is propagated outwards from the root  The identities of the nodes with the answer are propagated back to the root  The root establishes communication with one of the above nodes and receives the answer from it  In the third step all nodes along the path as well as the node with the answer receive their rewards

6. Nash Equilibrium  v (f,x) is the probability that the subtree below v possesses the answer given that v offers rewards x and v itself does not have the answer  v (f,x) = 1 -  v (f,x)  v (f,x) =  w is child of v [1-q(1-p  w (f,f w (x)))] Pay-off for node v = c 1 + c 2 (r-x-1)  v (g,x)  r is reward offered to v  x is the reward v offers to its children  g is Nash Equilibrium strategy if each g v in g maximizes the pay- off for node v, for all nodes v (Theorem 2.1)  g v is same for all nodes i.e. all nodes play the same strategy in the state of Nash Equilibrium If p generalizes q then the Nash Equilibrium is unique (Theorem 2.2)

7. Breakpoint Structure of Rewards R  (n,b): minimum utility required by root v* in order to obtain an answer with probability at least . Assume n>1 and b>1 are fixed  The set of possible values for  is partitioned into intervals  R  (n,b) is constant within each interval but increases at a ‘breakpoint’ between two intervals  If we increase utility r* at the node, nodes tend to push the reward deeper into the tree  However a change in the minimum utility R  (n,b) is observed only when this tendency to push, propagates the query to an extra level of depth in the tree  (r): Number of nodes the query would reach if the root had utility r, all nodes were active and no node possessed the answer i.e. the maximum possible level that a query can reach if the root has utility r.

8. Breakpoint Structure of Rewards In case of networks with no incentives  j probability that no node in the first j levels has the answer given that the root does not We have,  v* (g,r) =   (r) u j is minimum r for which  (r)>j-1 For a given initial utility r, the optimal reward root v* can offer to its children in order to maximize its pay-off is of the form u i for some i Pay-off for root having utility r and offering reward u i is given by l i (r)=(r-u i -1)(1-  i ) Suppose for all r >= u j, we have l j-1 (r) > l j-2 (r) > … > l 1 (r)  y j+1 is the point where l j intersects l j-1 and u j+1 = greatest_int(y j+1 )  We have, for all r >= u j+1, l j (r) > l j-1 (r) > … > l 1 (r) If  ’ j = y j – u j-1 and  j = u j – u j-1 then,

9. Growth Rate of Rewards Let function t(x) = (1-q(1-px)) and we have  j = t(  j-1 )

10. Growth Rate of Rewards (b<2) Choose  0 1 Consider sequence of  j values up to the point it drops below 1-  First segment of sequence of  j to be the set of indices j for which  j >= 1-  0 /n for  0 > b/(2-b) Second segment to be set of indices j for which 1-  0 /n >  j >= 1-  0

11. Growth Rate of Rewards (b<2)

12. Growth Rate of Rewards (b>2) Choose  0 2 Consider sequence of  j values up to the point it drops below 1-  First segment of sequence of  j ’s to be set if indices j for which  j >= 1-  0 Second segment to be set of indices j for which 1-  0 >  j >= 1- 

13. Growth Rate of Rewards (b>2)

14. Extensions and Future Directions Analysis of the neighborhood of b=2 Behavior of lower bound when b approaches 1 from above Incorporating more complexity in the model  More complex queries  Adding more factors such as response time Incentive Queries in Directed Acyclic Graphs and a Model of Competition