Inefficiency caused by Risk

Transcription

1 Inefficiency caused by Risk Aversion in Selfish Rou<ng Thanasis Lianeas Evdokia Nikolova University of Texas at Aus2n Nicolas S<er- Moses Facebook Schloss Dagstuhl Oct 2015

2 Speeds in network are stochas2c Sca<erplot speed vs. 2me of day Source: Arvind Thiagarajan, Paresh Malalur, CarTel.csail.mit.edu

3 Commuters pad travel 2mes Worst case > twice free flow 2me Source: Texas Transporta2on Ins2tute; ABC News Survey.

4 Goal of this Work Understand impact of risk- aversion on conges2on, by studying resul2ng traffic assignment: Uncertain delays influence users decisions Equilibrium existence, encoding, efficiency* Price of Risk Aversion** * E. Nikolova, N. Stier-Moses. SAGT 2011 / Operations Research 2014 ** E. Nikolova, N. Stier-Moses. EC 2015 ** T. Lianeas, E. Nikolova, N. Stier-Moses. Working paper 2015

5 Related Work Rou2ng Games: Wardrop 52, Beckmann et al. 56, A lot of work in Transporta2on, Algorithmic Game Theory, Opera2ons Research, Surveys in Nisan et al. 07, Correa & S.M. 11 Stochas2c Equilibrium models: Dial 71, Gupta, Stahl, Whinston 97 Risk- aversion in rou2ng games: a few references in transporta2on (but not too many). Ordóñez & S.M. 10, Nie 11, Angelidakis- Fotakis- Lianeas 13, Meir- Parkes 15.

6 Add error term to Wardrop model Directed graph G = (V,E) Single OD pair (s,t) and unit demand Nonatomic players (flow model) Strategy set: feasible s- t paths Players decisions: flow vector Edge delay func2ons: l e x R Ρaths ( x ) + ξ ( x ) e e e Expected delay Random variable with mean 0, st.dev. σ e )

7 Individual goal: minimum- risk paths Mean- variance path cost: Q path (x) = 2 l e ) + r σ e ) = le )+ rσ 2 e ) Pros e path e path e path Simple, well- known method to incorporate risk- aversion (e.g., finance) Addi2ve (path risk computed as sum of edge risks) Cons ( ) May result in stochas2cally dominated paths Not as meaningful as mean- standard devia2on objec2ve

8 Risk- averse vs Risk- neutral Wardrop Equilibrium Defini2on: A flow x is at equilibrium if for every OD pair k and for every path with posi2ve flow: Q ( x) Q '( x), for every path' path path Players select minimum paths wrt to func2on Q Risk- Averse Wardrop Equilibrium (RAWE): general case is mean- variance cost of a path Risk- Neutral Wardrop Equilibrium (RNWE): case r=0 is mean cost of path 2 Q path (x) = l e ) + r σ e ) = le )+ rσ 2 e ) e path e path ( ) e path

9 Are Risk- Averse Equilibria Efficient? POA: Impact of selfish behavior by comparing equilibrium to op2mum POA with risk- aversion: same with RAWE Theorem [Nikolova, S.M. 11]: POA with risk aversion = POA in classic conges2on games Problem: selfish behavior and risk aversion coupled together. Not clear which causes the inefficiency Decouple both by comparing to risk- neutral WE

10 Price of Risk Aversion Cost of Flow C(x): although users are risk- averse, central planner is risk- neutral. Consider the sum of expected travel Cmes Price of Risk Aversion (PRA): captures inefficiency introduced by user risk- aversion by comparing with the risk- neutral case r C(x ) sup 0 C(x ) problem instances Risk- averse WE (RAWE) Risk- neutral WE (RNWE)

11 Risk- averse vs Risk- neutral equilibria Example: Send one unit of flow from S to T mean (1+rk)x, var 0 S x 1 - x mean 1, var k T Risk- averse WE: Route all flow on top; cost 1+rk Risk- neutral WE: Route flow on both links; cost 1 Price of risk aversion: 1+rk

12 Price of Risk Aversion Price of Risk Aversion is unbounded in general, but uncertainty is not arbitrary in real world Consider a bounded variance- to- mean ra2o: σ 2 e )/l e ) k GOAL: Compute PRA for fixed k Restricted topology but general latency func2ons General topology but restricted latency func2ons

13 Price of Risk Aversion: Upper Bound for Arbitrary Latency Func2ons Theorem: In a general graph, PRA 1+ηrk Here, η is a graph topology parameter: # forward subpaths in an alterna2ng path [ η ½ V ] Intui2on: For Pigou networks: For series- parallel networks: For Braess networks: For domino- free networks: PRA 1+1rk PRA 1+1rk PRA 1+2rk PRA 1+2rk

14 Price of Risk Aversion: Upper Bound for Arbitrary Latency Func2ons Theorem: In a general graph, PRA 1+ηrk Here, η is a graph topology parameter: # forward subpaths in an alterna2ng path [ η ½ V ] Proof idea: Compare equilibria on an alterna2ng path: forward edges have higher RNWE flow, and backward edges have higher RAWE flow

15 Price of Risk Aversion in a general graph Theorem: In a general graph, PRA 1+ηrk Proof sketch: Lemma 1: There is an alterna2ng path s.t. forward edges have higher RNWE flow, and backward edges have higher RAWE flow Lemma 2: cost C(x) of RAWE x sa2sfies: C(x) (1+ kr) l e ) l e ) e forward Lemma 3: cost C(z) of RNWE z sa2sfies: e backward l e (z e ) l e (z e ) C(z) e forward e backward

16 Price of Risk Aversion in a general graph Theorem: In a general graph, PRA 1+ηrk Proof: C(x) (1+ kr) l e ) l e ) e forward e backward (1+ kr) l e (z e ) l e (z e ) e forward e forward C(z)+ kr l e (z e ) C(z)+ krηc(z) e backward by Lemma 2 by Lemma 1 by Lemma 3 by defini2on of RNWE z

17 Price of Risk Aversion: Lower Bound for Arbitrary Latency Func2ons Theorem: In a general graph, PRA 1+ηrk

18 Price of Risk Aversion: Upper Bound for Fixed Latency Func2ons Theorem: Considering (1,μ)- smooth cost func2ons and arbitrary graphs, PRA = (1+rk)(1- μ) - 1 [VI approach of Correa, Schulz, S.M. 04] Proof idea: For x=rawe, and z=rnwe: Par22on arcs between A={e x e < z e } and B={e x e z e }, use VI: and smoothness property: zl(x) zl(z) + μxl(x)

19 Summary & open ques2ons Tight bounds for effects of risk- aversion in rou2ng For general graphs and mean, variance func2ons: C(RAWE) (1+ηrk) C(RNWE) where η=1 for series- parallel graphs, η=2 for Braess graph, η V /2 for a general graph For general graphs and μ- smooth latency func2ons: C(RAWE) (1+rk) (1- μ) - 1 C(RNWE) Does it extend to mean- stdev risk? Heterogeneous risk a tudes; other risk func2ons?