Query Incentive Networks

Jon Kleinberg and Prabhakar Raghavan

- Presented by: Nishith Pathak

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

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

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

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

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-pw(f,fw(x)))] Pay-off for node v = c1 + c2(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 gv in 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 p generalizes q then the Nash Equilibrium is unique (Theorem

2.2)

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.

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)

uj 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 ui for some i Pay-off for root having utility r and offering reward ui is given by

li(r)=(r-ui-1)(1-i) 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 ’j = yj – uj-1 and j = uj – uj-1 then,

Growth Rate of Rewards

Let function t(x) = (1-q(1-px)) and we havej = t(j-1)

Growth Rate of Rewards (b<2) Choose 0 < and n large enough such that

pb(1-2bd0)>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

Growth Rate of Rewards (b<2)

Growth Rate of Rewards (b>2) Choose 0 < and n large enough such that

pb(1-2bd0)>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-

Growth Rate of Rewards (b>2)

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