Near Optimal Network Design With Selfish Agents Eliot

Near Optimal Network Design With Selfish Agents Eliot Anshelevich Anirban Dasupta Eva Tardos Tom Wexler Cornell University Presented by: Andrey Stolyarenko School of CS, Tel-Aviv University Some of the slides are taken from E. Anshelevich and L. Kaiser presentations

Selfish Agents in Networks l Traditional network design problems are centrally controlled l What if network is instead built by many self-interested agents? l As we saw on previous lectures, properties of resulting network may be very different from the globally optimum one

Connection Games

The Connection Game – A Story Think of sea transport companies or broadband internet providers. These are our agents l each company needs to connect a few ports or users l every connection has a constant cost l connection is bought if all together pay for it

The Connection Game – Selfish as usual l We do not consider negotiations, communication l No external mechanism or regulation l All desired users must be connected, no tradeoff l Everyone will go for a cheaper price if possible

The Connection Game – Model

The Connection Game – Example s 1 t 3 s 2 t 2 s 3 t 1

Sharing Edge Costs l How should multiple players on a single edge split costs? One approach: no restrictions. . . any division of cost agreed upon by players is OK. l TODAY Near-Optimal Network Design with Selfish Agents STOC ‘ 03 Anshelevich, Dasgupta, Tardos, Wexler. l Another approach: try to ensure some sort of fairness. NEXT WEEK… The Price of Stability for Network Design with Fair Cost Allocation FOCS ’ 04 Anshelevich, Dasgupta, Kleinberg, Tardos, Wexler, Roughgarden

? What are we interested in l From Nash’s Theorem (1950) we know that mixed-strategy (non deterministic) Nash Equilibria always exist l There for We are interested in purestrategy (deterministic) Nash Equilibria l From now and on “Nash Equilibria” (“NE”) will mean: l Deterministic Nash Equealibira

? What are we interested in l How bad can NE be? – Price of Anarchy l How good can NE be? – Price of Stability l (1+ e)-approx. NE

Nash Equilibrium t 2 A NE is a set of payments for players such that no player wants to deviate. l A player must connect his terminals t 1 s 3 A player does not care whether other players connect. l When considering deviations, s 1 a player assumes that other players’ payments are fixed. l t 3 s 2

Nash Equilibria - Formal

Three Observations

Example 1 - Two Different NEs t 1 , t 2 , … tk t t 1 k s s 1 , s 2 , … sk 1 t k s One NE: each player pays 1/k 1 k s Another NE: each player pays 1

Reminder: The POA and POS Price of Anarchy = [Koutsoupias, Papadimitriou] [Roughgarden, Tardos] cost(worst NE) s 1…sk cost(OPT) (Min cost Steiner forest) 1 Price of Stability = k cost(best NE) cost(OPT) t 1…tk Question: What were the POA and POS in Example 1 ?

NE Doesn’t have to Exits! Don’t forget NE=pure-NE for now

Example 2 - No Nash s 1 all edges cost 1 a d s 2 t 2 b c t 1

Example 2 - No Nash s 1 all edges cost 1 a d s 2 t 2 b c t 1 We know that any NE must be a tree: WLOG assume the tree is a, b, c.

Example 2 - No Nash s 1 all edges cost 1 a d s 2 t 2 b c t 1 We know that any NE must be a tree: WLOG assume the tree is a, b, c. Only player 1 can contribute to a. Only player 2 can contribute to c.

Example 2 - No Nash s 1 all edges cost 1 a d s 2 t 2 b c t 1 We know that any NE must be a tree: WLOG assume the tree is a, b, c. Only player 1 can contribute to a. Only player 2 can contribute to c. Neither player can contribute to b, since d is a tempting deviation.

When NE exist, how bad can it be? l In The Connection Game the POA is at most N - The number of agents l If the worst NE p const more than N times OPT then there must be a player i whose payments pi are strictly more then OPT l Player i could deviate by purchasing the entire optimal solution by himself

? When NE exist, how good can it be l In Exaple 1 we saw that POS was 1 NEXT!

Single Source Games

Simple Case - MST Easy if all nodes are terminals: Players buy edge above them in OPT. Claim: This is a Nash Equilibrium. ( i unhappy => can build cheaper tree ) • Typically we will have Steiner nodes. Who buys the edge above these?

Attempts to Buy Edges 1) Can we get a single player to pay? Both players must help buy top edge. 3 5 3 2) Can we split edge costs evenly? 4 4 4 5 4 Second node won’t pay more than 5 in total.

Greedy Algorithm In both examples, players were limited by possible deviations. e Given OPT, pay for edges in OPT from the bottom up, greedily (openhanded) , as constrained by deviations. If we buy all edges, we’re done!

Single Source Games

Notation e

The Greedy Algorithm

Example 4 4 3 5 4 4 4 5 4 3 5 3

!We get NE If we buy all edges we are done!

Proof Idea If greedy fails to pay for e, we will show that the tree is not OPT. l All players have possible deviations. l Deviations and current payments must be equal. l If all players deviate, all connect, but pay less. l e

Proof Proof

Path Lemma

Proof Finale e

!But, Wait Suppose greedy algorithm cannot pay for e e e’ 1 4 2 3 Further, suppose 1 & 2 share cost(e’) l Consider 1 & 2 both deviating… l Player 1 stops contributing to e’ l Danger: 2 still needs this edge! l

Don’t Worry, Everything is fine. Just, e e’ 1 2 3 4 Shouldn’t allow player 1 to deviate: If only 2 deviates, all players reach the source. Idea: should use the “highest” deviating paths first.

(1+ e)-approx. NE in Polytime Theorem: For single source, can find a (1+ε)-approx. NE in polytime on an α-approx. Steiner tree. α = best Steiner tree approx. (1. 55) ε > 0, running time depends on ε. Proof Sketch: • Greedy algorithm from previous proof either finds a NE or a cheaper tree than it was given. • Only take significant improvements.

Multi Source Games

Price of Anarchy in Multi-Source Games s 1 ε O(k) ε s 2 t 2 ε ε t 1 O(k) s 3…sk 1 t 3…tk OPT costs ~1, but it’s not a NE. The only NE costs O(k), so optimistic price of anarchy is almost k.

Result for Multi Source Games We know a NE may not exist, so settle for approximate NE. How bad an approximation must we have if we insist on buying OPT? 2 3 1 1 2 3 Theorem: For any game, there exists a 3 -approx NE that buys OPT. Note: this is true even for games where players may have more than 2 terminals.

Proof Idea • Break up OPT into chunks. • Use optimality of OPT to show that any player buying a single chunk has no incentive to deviate. 2 3 1 • Each chunk is paid for by a single player. • Each player pays for at most 3 chunks. 1 2 3

Connection Sets 1 l A connection set C of player i is a set of edges such that: ¡ ¡ l C only includes edges on the path Pi from si to ti in OPT. If OPT is bought, and i pays only for C, then i has no incentive to deviate. b a Connection set = chunk 1

Main Challenge Form a payment scheme where each player pays for at most 3 connection sets. l i pays for edges that no other players would pay for in OPT. l Another connection set for each terminal of i. l 2 3 1 2 1 3

Tree Decomposition l Decompose OPT into hierarchical paths, where each path begins at a terminal and ends at a path of higher level. 4 1 3 2 2 3 1 5 4 5

Payment Scheme l Connection sets in each path P are paid for by terminals associated with paths entering P. 2 2 1 3 4

Payment Scheme l Connection sets in each path P are paid for by terminals associated with paths entering P. 4 3 1 3 2 5 2 3 2 1 1 5 4 5

Approximation Algorithm Theorem: For multi-source 2 -terminal games, can find a (3+ε)-approx. NE in polytime on an 1. 55 -approx. to OPT. For >2 terminals, above approximation becomes (4. 65+ε), since need to use best known approx for Steiner tree.

Results and More l Single Source POS = 1 ¡ Polytime NE approx ¡ What happens in directed graphs? ¡ What happens if we add a maximum payment that a player is willing to may in order to stay connected? ¡

Results and More l Multi Source The existence of NE is NPC if the number of players is a part of the input. Show by 3 -SAT reduction ¡ POS can be O(n) ¡ (3+ε)-NE approx. always exist ¡ (4. 65+ε)-NE approx algorithm for 1. 55 OPT ¡ There are games which the best NE is 1. 5 approx. Lower bound is 1. 5. ¡

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