Query Incentive Networks

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-eq,d>0

5. Formulating a Model • How much utility r* is required by the root node v* in order to achieve a probability s 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 • av(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 • bv(f,x) = 1 - av(f,x) • bv(f,x) = Pw is child of v[1-q(1-pbw(f,fw(x)))] • Pay-off for node v = c1 + c2(r-x-1)av(g,x) • r is reward offered to v • x is the reward v offers to its children • g is Nash Equilibrium strategy if each gvin g maximizes the pay-off for node v, for all nodes v(Theorem 2.1) • gv is same for all nodes i.e. all nodes play the same strategy in the state of Nash Equilibrium • If pgeneralizesq then the Nash Equilibrium is unique (Theorem 2.2)

7. Breakpoint Structure of Rewards • Rs(n,b): minimum utility required by root v* in order to obtain an answer with probability at least s. • Assume n>1 and b>1 are fixed • The set of possible values for s is partitioned into intervals • Rs(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 Rs(n,b) is observed only when this tendency to push, propagates the query to an extra level of depth in the tree • d(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 fj probability that no node in the first j levels has the answer given that the root does not We have,bv*(g,r) = fd(r) • uj is minimum r for which d(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 ui for some i • Pay-off for root having utility r and offering reward ui is given by li(r)=(r-ui-1)(1-fi) • Suppose for all r >= uj, we have lj-1(r) > lj-2(r) > … > l1(r) • yj+1 is the point where lj intersects lj-1 and uj+1 = greatest_int(yj+1) • We have, for all r >= uj+1, lj(r) > lj-1(r) > … > l1(r) • If D’j = yj – uj-1 and Dj = uj – uj-1 then,

9. Growth Rate of Rewards • Let function t(x) = (1-q(1-px)) and we havefj = t(fj-1)

10. Growth Rate of Rewards (b<2) • Choose s0 < sand n large enough such that pb(1-2bds0)>1 • Consider sequence of fj values up to the point it drops below 1-s • First segment of sequence of fjto be the set of indices j for which fj >= 1-k0/n for k0 > b/(2-b) • Second segment to be set of indices j for which 1-k0/n > fj >= 1-s0

11. Growth Rate of Rewards (b<2)

12. Growth Rate of Rewards (b>2) • Choose s0 < s and n large enough such that pb(1-2bds0)>2 • Consider sequence of fj values up to the point it drops below 1-s • First segment of sequence of fj’s to be set if indices j for which fj >= 1-s0 • Second segment to be set of indices j for which 1-s0 > fj >= 1-s

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