algorithms, game theory and risk-averse decision making*...shortest paths example algorithms, game...
TRANSCRIPT
EvdokiaNikolovaUniversityofTexasatAus2n
Algorithms,GameTheoryandRisk-averseDecision
Making*
*IntroductorylectureforSimonsReal-BmeDecisionMakingBootcamp,January24,2018
Real-2medecisionmakingexamples
Algorithms,GameTheory&Risk-averseDecisionMakingEvdokiaNikolova
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Real-2medecisionmakingexamples
Algorithms,GameTheory&Risk-averseDecisionMakingEvdokiaNikolova
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PartI:Algorithms
EvdokiaNikolova Algorithms,GameTheory&Risk-averseDecisionMaking
Algorithms:thebasics
Algorithms,GameTheory&Risk-averseDecisionMaking
• What is an algorithm? – a step-by-step
procedure to solve a problem
– every computer program is the instantiation of some algorithm
EvdokiaNikolova
Shortestpathsexample
Algorithms,GameTheory&Risk-averseDecisionMaking
• Solves a general, well-specified problem – given a graph G = (V, E), a source node s and a destination node t,
and edge costs c1,…,cn, as input, produce as output a shortest st-path, namely an st-path with smallest total edge cost
• Problem has specific instances
• Algorithm takes every possible instance and produces output with desired properties – Dijkstra, Bellman-Ford, Floyd-Warshall shortest path algorithms
EvdokiaNikolova
Modelingthereal-world
Algorithms,GameTheory&Risk-averseDecisionMaking
• Cast your application in terms of well-studied abstract data structures
EvdokiaNikolova
Concrete Abstract
arrangement, tour, ordering, sequence permutation
cluster, collection, committee, group, packaging, selection subsets
hierarchy, ancestor/descendants, taxonomy, radial network trees
network, circuit, web, relationship graph
sites, positions, locations points
shapes, regions, boundaries polygons
text, characters, patterns strings
Graphterminology
Algorithms,GameTheory&Risk-averseDecisionMaking
• Adirectedgraph(ordigraph)Gisapair(V,E)whereVisafiniteset(of“ver2ces”)andE(the“edges”)isasubsetofV×V.
• AnundirectedgraphGisapair(V,E)whereVisafiniteset(of
“ver2ces”)andE(the“edges”)isasetofunorderedpairsofedges{u,v},whereu≠v.
EvdokiaNikolova
1 2
36
1 2 3 41 2 3
45 1 2 3 4
5 61 2 3 4
5 64
1 2 3
u v w x y z
5 61 2 3 4
5 64
1 2 3 4
1 2
36
1 2 3 41 2 3
45 1 2 3 4
5 61 2 3 4
5 64
1 2 3
u v w x y z
5 61 2 3 4
5 64
1 2 3 4
Graph=NetworkNodes=Ver2cesEdges=Links=Arcs
• Abstrac2onoftransporta2ongraph(Braessparadoxgraph)
B A
Graphexamples
Algorithms,GameTheory&Risk-averseDecisionMakingEvdokiaNikolova
0.5hr 1 hour
1 hour 0.5hr
1hr
1hr
0
• Abstrac2onoftransporta2ongraph(Braessparadoxgraph)
• Abstrac2onofelectricitygraph:
B A
Graphexamples
Algorithms,GameTheory&Risk-averseDecisionMakingEvdokiaNikolova
0.5hr 1 hour
1 hour 0.5hr
1hr
1hr
0
Bus=NodeLine=Edge?
B A
Graphexamples
Algorithms,GameTheory&Risk-averseDecisionMaking
• Abstrac2onoftransporta2ongraph(Braessparadoxgraph)
• Abstrac2onofelectricitygraph:
EvdokiaNikolova
0.5hr 1 hour
1 hour 0.5hr
1hr
1hr
0
Bus=NodeLine=Edge?
EvdokiaNikolova Algorithms,GameTheory&Risk-averseDecisionMaking
Backtoshortestpaths• Algorithmicchallenge:exponen2allymanypaths
• Bruteforce:enumerateallpossiblepaths;outputbestone.Bruteforcealgorithmhasexponen2alrunning2me
V1 V2 V3 Vn-1 Vn ...5
3
2
3
5
4
…
2n
EvdokiaNikolova Algorithms,GameTheory&Risk-averseDecisionMaking
Backtoshortestpaths• Algorithmicchallenge:exponen2allymanypaths
• Bruteforce:enumerateallpossiblepaths;outputbestone.Bruteforcealgorithmhasexponen2alrunning2me
V1 V2 V3 Vn-1 Vn ...5
3
2
3
5
4
…
2n1067:numberofatomsinourgalaxy
EvdokiaNikolova Algorithms,GameTheory&Risk-averseDecisionMaking
Backtoshortestpaths• Algorithmicchallenge:exponen2allymanypaths
• Bruteforce:enumerateallpossiblepaths;outputbestone.Bruteforcealgorithmhasexponen2alrunning2me
V1 V2 V3 Vn-1 Vn ...5
3
2
3
5
4
…
2n
EvdokiaNikolova Algorithms,GameTheory&Risk-averseDecisionMaking
Backtoshortestpaths• Algorithmicchallenge:exponen2allymanypaths
• Inshortestpaths,op*malsubstructureproperty(partoftheop*malsolu*onremainsop*malonthesubproblem)allowsforefficientdynamicprogrammingalgorithms(Dijkstra,Bellman-Ford,etc)
AB
C
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6.9
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V1 V2 V3 Vn-1 Vn ...5
3
2
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Dijkstrashortestpathalgorithm
EvdokiaNikolova Algorithms,GameTheory&Risk-averseDecisionMaking
B A
0.5hr
0.5hr
1hr
1hr
0
inf
inf
inf
Dijkstrashortestpathalgorithm
EvdokiaNikolova Algorithms,GameTheory&Risk-averseDecisionMaking
B A
0.5hr
0.5hr
1hr
1hr
0
0.5
1
inf
Dijkstrashortestpathalgorithm
EvdokiaNikolova Algorithms,GameTheory&Risk-averseDecisionMaking
B A
0.5hr
0.5hr
1hr
1hr
0
0.5
0.5
1.5
Dijkstrashortestpathalgorithm
EvdokiaNikolova Algorithms,GameTheory&Risk-averseDecisionMaking
B A
0.5hr
0.5hr
1hr
1hr
0
0.5
0.5
1
Dijkstrashortestpathalgorithm
EvdokiaNikolova Algorithms,GameTheory&Risk-averseDecisionMaking
B A
0.5hr
0.5hr
1hr
1hr
0
0.5
0.5
1
(Basic)AlgorithmDesignTechniques
Algorithms,GameTheory&Risk-averseDecisionMaking
• Brute Force & Exhaustive Search – follow definition / try
all possibilities • Divide & Conquer
– break problem into distinct subproblems
• Transformation – convert problem to
another one
EvdokiaNikolova
• Dynamic Programming – break problem into
overlapping subproblems
• Greedy – repeatedly do what is
best now • Randomization
– use random numbers • Linear programming
EvdokiaNikolova Algorithms,GameTheory&Risk-averseDecisionMaking
Algorithmrunning2me
• Wewouldlikeadefini2onofalgorithmefficiencythatis:– plaiorm-independent– instance-independent– ofpredic2vevaluewithrespecttoincreasinginstance(input)sizes(e.g.,foragraph,inputsizeisthenumberofnodesnandedgesm)
• AnalgorithmisefficientifithaspolynomialrunningBme.– f(n)=aknk+ak-1nk-1+…+a1n+a0isapolynomialofdegreek.
EvdokiaNikolova Algorithms,GameTheory&Risk-averseDecisionMaking
Algorithmrunning2me• Analgorithmisefficientifithaspolynomialrunning
Bme.– f(n)=aknk+ak-1nk-1+…+a1n+a0isapolynomialofdegreek.– Wecareabouthighestordertermandnocoefficients:algorithmpolynomialrun2meisO(nk),wherekisaconstant
• Ofcourse,running2meofn100isclearlynotgreat,and
arunning2meofn1+.02(logn)isnotclearlybad.Butinprac2ce,polynomial2meisgenerallygood.
• Inaddi2ontobeingprecise,thisdefini2onisalsonegatable.
PvsNP
Algorithms,GameTheory&Risk-averseDecisionMaking
• P is the class of problems which can be solved in polynomial time
• NP (“nondeterministic polynomial time”) is the class of problems for which a candidate solution can be verified in polynomial time – Note: NP does not stand for “not polynomial” – P is a subset of NP
EvdokiaNikolova
P
NP
PvsNP
Algorithms,GameTheory&Risk-averseDecisionMaking
Difference between solving a problem and verifying a candidate solution: • Solving a problem: is there a path in graph G from vertex u
to vertex v with at most k edges? • Verifying a candidate solution: is v0, v1, …, vl a path in
graph G from vertex u to vertex v with at most k edges?
EvdokiaNikolova
PvsNP
Algorithms,GameTheory&Risk-averseDecisionMaking
Difference between solving a problem and verifying a candidate solution: • Solving a problem: is there a path in graph G from vertex u
to vertex v with at most k edges? • Verifying a candidate solution: is v0, v1, …, vl a path in
graph G from vertex u to vertex v with at most k edges?
EvdokiaNikolova
Isthereapathoflengthatmostk=5?
source
des2na2on
PvsNP
Algorithms,GameTheory&Risk-averseDecisionMaking
Difference between solving a problem and verifying a candidate solution: • Solving a problem: is there a path in graph G from vertex u
to vertex v with at most k edges? • Verifying a candidate solution: is v0, v1, …, vl a path in
graph G from vertex u to vertex v with at most k edges?
EvdokiaNikolova
Isthereapathoflengthatmostk=5?
source
des2na2on
PvsNP
Algorithms,GameTheory&Risk-averseDecisionMaking
• A Hamiltonian cycle in an undirected graph is a cycle that visits every vertex exactly once.
• Solving a problem: is there a Hamiltonian cycle in graph G?
• Verifying a candidate solution: is v0, v1, …, vl a Hamiltonian cycle of graph G?
EvdokiaNikolova
PvsNP
Algorithms,GameTheory&Risk-averseDecisionMaking
• A Hamiltonian cycle in an undirected graph is a cycle that visits every vertex exactly once.
• Solving a problem: is there a Hamiltonian cycle in graph G?
• Verifying a candidate solution: is v0, v1, …, vl a Hamiltonian cycle of graph G?
EvdokiaNikolova
PvsNP
Algorithms,GameTheory&Risk-averseDecisionMaking
• Although poly time verifiability seems like a weaker condition than poly time solvability, no one has been able to prove that it is weaker (describes a larger class of problems)
• So it is unknown whether P = NP.
EvdokiaNikolova
all problems
P �
NP �or
all problems
P=NP � ?
NP-Completeproblems
Algorithms,GameTheory&Risk-averseDecisionMaking
• NP-complete problems is class of "hardest" problems in NP. • They have the property that if any NP-complete problem
can be solved in poly time, then all problems in NP can be, and thus P = NP.
EvdokiaNikolova
?all problems
P �
NP �or
all problems
P=NP=NPC �NPC �
NPC = NP-complete
NP-Completeproblems
Algorithms,GameTheory&Risk-averseDecisionMaking
• NP-complete problems is class of "hardest" problems in NP. • A decision problem is NP-complete if
– It is in NP – It is “at least as hard as” some known NP-complete problem
EvdokiaNikolova
…
• When given a new problem, a computer science theorist first wants to understand whether the problem is in P, or NP-complete, or another complexity class.
FirstNP-CompleteProblem(SAT) Problem2 Problem3 Problemk
OurProblem
NP-Completeproblems
Algorithms,GameTheory&Risk-averseDecisionMaking
• NP-complete problems is class of "hardest" problems in NP. • They have the property that if any NP-complete problem
can be solved in poly time, then all problems in NP can be, and thus P = NP.
• Other complexity classes:
EvdokiaNikolova
PvsNPproblem
Algorithms,GameTheory&Risk-averseDecisionMaking
• Open question since about 1970 • One of the seven"MillenniumPrizeProblems” bytheClayMathema2csIns2tute($1millionprizeforsolvingit)–alongwiththeRiemannHypothesis,thePoincaréConjecture,etc.*
• Great theoretical interest • Great practical importance:
– If your problem is NP-complete, then don't waste time looking for an efficient algorithm
– Instead look for efficient approximations, heuristics, etc.
EvdokiaNikolova
*htp://www.claymath.org/millennium-problems
PvsNPproblem
Algorithms,GameTheory&Risk-averseDecisionMaking
• Open question since about 1970 • One of the seven"MillenniumPrizeProblems” bytheClayMathema2csIns2tute($1millionprizeforsolvingit)–alongwiththeRiemannHypothesis,thePoincaréConjecture,etc.*
• Great theoretical interest • Great practical importance:
– If your problem is NP-complete, then don't waste time looking for an efficient algorithm
– Instead look for efficient approximations, heuristics, etc.
EvdokiaNikolova
*htp://www.claymath.org/millennium-problems
PvsNPproblem
Algorithms,GameTheory&Risk-averseDecisionMaking
• Open question since about 1970 • One of the seven"MillenniumPrizeProblems” bytheClayMathema2csIns2tute($1millionprizeforsolvingit)*
• Great theoretical interest • Great practical importance:
– If your problem is NP-complete, then don't waste time looking for an efficient algorithm
– Instead look for efficient approximations, heuristics, etc.
EvdokiaNikolova
*htp://www.claymath.org/millennium-problems
Approxima2onalgorithms*
Algorithms,GameTheory&Risk-averseDecisionMaking
• If we do not know how to solve a problem in polynomial time, can we find a near-optimal solution in polynomial time?
• An α-approximation algorithm for an optimization problem – runs in polynomial time – always returns a candidate solution – cost of returned solution is at most α times the cost
of the optimal solution (for a minimization problem)
EvdokiaNikolova*Thissemester:CS294-145"Approxima2onAlgorithms"instructorDavidWilliamson.Tue/Thu3:30-5:00in310SodaHall.
TravelingSalesmanProblem
Algorithms,GameTheory&Risk-averseDecisionMaking
• Given a set of cities, distances between all pairs of cities, and a bound B, does there exist a tour (sequence of cities to visit that returns to the start and visits each city exactly once) that requires at most distance B to be traveled?
• TSP is in NP: – given a candidate solution (a tour), add up all
the distances and check if total is at most B
EvdokiaNikolova
TSPapproxima2onalgorithm
Algorithms,GameTheory&Risk-averseDecisionMaking
• Input: set of cities and distances b/w them that satisfy the triangle inequality
EvdokiaNikolova
EvdokiaNikolova Algorithms,GameTheory&Risk-averseDecisionMaking
A
B C
D
E
F G
TSPapproxima2onalgorithm
1) Compute MST
• Input: set of cities and distances b/w them that satisfy the triangle inequality
EvdokiaNikolova Algorithms,GameTheory&Risk-averseDecisionMaking
A
B C
D
E
F G
TSPapproxima2onalgorithm
1) Compute MST 2) Go around MST to
get a tour
• Input: set of cities and distances b/w them that satisfy the triangle inequality
EvdokiaNikolova Algorithms,GameTheory&Risk-averseDecisionMaking
A
B C
D
E
F G
TSPapproxima2onalgorithm
1) Compute MST 2) Go around MST to
get a tour 3) Remove duplicate
cities
• Input: set of cities and distances b/w them that satisfy the triangle inequality
EvdokiaNikolova Algorithms,GameTheory&Risk-averseDecisionMaking
A
B C
D
E
F G
TSPapproxima2onalgorithm
• This is a 2-approximation algorithm
1) Compute MST 2) Go around MST to
get a tour 3) Remove duplicate
cities
• Input: set of cities and distances b/w them that satisfy the triangle inequality
PartII:GameTheory
EvdokiaNikolova Algorithms,GameTheory&Risk-averseDecisionMaking
Best route depends on others
Algorithms,GameTheory&Risk-averseDecisionMakingEvdokiaNikolova
• Gamesarethoughtexperimentstohelpuslearnhowtopredictra)onalbehaviorinsitua)onsofconflict
• Situa)onofconflict:Everybody'sac2onsaffectothers.Thisiscapturedbythetabulargameformalism.
• Ra)onalBehavior:Theplayerswanttomaximizetheirownexpectedu2lity.Noaltruism,envy,masochism,orexternali2es(ifmyneighborgetsthemoney,hewillbuylouderstereo,soIwillhurtalitlemyself...).
• Predict:Wewanttoknowwhathappensinagame.Suchpredic2onsarecalledsolu2onconcepts(e.g.,Nashequilibrium).
Gametheory
Algorithms,GameTheory&Risk-averseDecisionMakingEvdokiaNikolova
Travel time increases with congestion
Algorithms,GameTheory&Risk-averseDecisionMaking
• Highwayconges2oncostswere$160billionin2014(TTI)
• Avg.commutertravels100minutesaday.EvdokiaNikolova
Example:Inefficiencyofequilibria
Suppose 100 drivers leave from town A towards town B.
What is the traffic on the network?
Every driver wants to minimize her own travel time.
In any unbalanced traffic pattern, all drivers on the most loaded path have incentive to switch their path.
Delay is 1.5 hours for everybody at the unique Nash equilibrium
Town A Town B
x/100 hours
1 hour
1 hour
1/2
1/2
x/100hours
x/100hours
1hour
1hour
Algorithms,GameTheory&Risk-averseDecisionMakingEvdokiaNikolova
Example:Inefficiencyofequilibria
A benevolent mayor builds a superhighway connecting the fast highways of the network.
What is now the traffic on the network?
No matter what the other drivers are doing it is always better for me to follow the zig-zag path.
Delay is 2 hours for everybody at the unique Nash equilibrium
Town A Town B
x/100 hours
x/100 hours
1 hour
1 hour
1
x/100hours
x/100hours
1hour
1hour
0hours
Algorithms,GameTheory&Risk-averseDecisionMakingEvdokiaNikolova
Example:Inefficiencyofequilibria
A B
x/100 hours
x/100 hours
1 hour
1 hour
1
A B
x/100 hours
x/100 hours
1 hour
1 hour
1/2
1/2
vs
Adding a fast road on a road-network is not always a good idea! Braess’s paradox
In the RHS network there exists a traffic pattern where all players have delay 1.5 hours.
PoA =performance of system in worst Nash equilibrium
optimal performance if drivers did not decide on their own
4/3
Price of Anarchy:
x/100hours
x/100hours
1hour
1hour
x/100hours
x/100hours
1hour
1hour
Algorithms,GameTheory&Risk-averseDecisionMaking
measures the loss in system performance due to free-will
EvdokiaNikolova
Equilibrium
• “Travel2mesonusedroutesareequalandnogreaterthantravel2mesonunusedroutes.”
• DefinesWardropEquilibrium,also called User Equilibrium or Nash Equilibrium.
Algorithms,GameTheory&Risk-averseDecisionMakingEvdokiaNikolova
0hoursTown A Town B
x/100 hours
x/100 hours
1 hour
1 hour
1
x/100hours
x/100hours
1hour
1hour
• “Theaverage[total]journey2meisminimum.”
• DefinesSocialOp2mum(SO)
SocialOp2mum
Algorithms,GameTheory&Risk-averseDecisionMakingEvdokiaNikolova
Town A Town B
x/100 hours
1 hour
1 hour
1/2
1/2
x/100hours
x/100hours
1hour
1hour
0hours
• “[Modified]Travel2mesonusedroutesareequalandnogreaterthantravel2mesonunusedroutes.”
• DefinesSocialOp2mum(SO)
SocialOp2mum
Algorithms,GameTheory&Risk-averseDecisionMakingEvdokiaNikolova
Town A Town B
x/100 hours
1 hour
1 hour
1/2
1/2
x/100hours
x/100hours
1hour
1hour
0hours
• “[Modified]Travel2mesonusedroutesareequalandnogreaterthantravel2mesonunusedroutes.”
• SocialOp2mum(SO)isEquilibriumw.r.ttravel2mesplustolls!
SocialOp2mum
Algorithms,GameTheory&Risk-averseDecisionMakingEvdokiaNikolova
Town A Town B
x/100 hours
1 hour
1 hour
1/2
1/2
x/100hours+1/2
x/100hours+1/2
1hour
1hour
0hours
• “[Modified]Travel2mesonusedroutesareequalandnogreaterthantravel2mesonunusedroutes.”
• SocialOp2mum(SO)isEquilibriumw.r.ttravel2mesplustolls!àMechanismdesign
SocialOp2mum
Algorithms,GameTheory&Risk-averseDecisionMakingEvdokiaNikolova
Town A Town B
x/100 hours
1 hour
1 hour
1/2
1/2
x/100hours+1/2
x/100hours+1/2
1hour
1hour
0hours
Note:bothEquilibriumandSocialop2mumexistandcanbecomputedefficiently.
PriceofAnarchy • Price of anarchy: (Koutsoupias, Papadimitriou ’99)
• Measures the degradation of system performance due to free will (selfish behavior)
• 4/3 in general graphs, linear delays as function of traffic; 2 for quartic delays (Roughgarden, Tardos ’02; Correa, Schulz, Stier-Moses ‘04, ‘08)
Cost Optimum SocialCost mEquilibriusup
instancesproblem
Algorithms,GameTheory&Risk-averseDecisionMakingEvdokiaNikolova
Take-awaypointsonconges2ongames
• Shortest paths (and algorithmic questions in general) can get complicated due to presence of more people
• Equilibrium and Social Optimum in nonatomic routing games exist and can be found efficiently via convex programs.
• Social optimum is an equilibrium with respect to modified latencies = original latencies plus toll.
• Price of anarchy: 4/3 for linear delays, can be found
similarly for more general classes of delay functions.
Algorithms,GameTheory&Risk-averseDecisionMakingEvdokiaNikolova
PartIII:Risk-aversedecisionmaking
EvdokiaNikolova Algorithms,GameTheory&Risk-averseDecisionMaking
EvdokiaNikolova Reliablerouteplanning
• Op2malroutemaydifferwithstart2me
Op2malroute?
30or60
40
EvdokiaNikolova Reliablerouteplanning
• Op2malroutemaydifferwithstart2me
Op2malroute?
30or60
40
10:30 am 9:50
late ontime
Departure time
EvdokiaNikolova Reliablerouteplanning
• Op2malroutemaydifferwithstart2me
Op2malroute?
30or60
40
10:30 am 9:50 10
late 50%
ontime
9:30 Departure time
• Op2malroutemaydifferwithstart2me
EvdokiaNikolova Reliablerouteplanning
Op2malroute?
30or60
40
10:30 am 9:50 10
Optimal route
Departure time
Implica2onsofriska{tudes
Algorithms,GameTheory&Risk-averseDecisionMaking
• Uncertaintyandriskcanaffectourdesignandanalysisof:
• Algorithms• AlgorithmicGameTheory• AlgorithmicMechanismDesign
EvdokiaNikolova
Whatisrisk?
Threemainquan2ta2veapproachestodefiningrisk:• Economics:Expectedu2litytheoryandalterna2ves• Finance:Markowitzmean-varianceframework• Modernrisktheory:axioma2capproachtodefiningrisk(coherent&convexriskmeasures)
• Systemicrisk:willnottalkaboutittoday.See“ASurveyofSystemicRiskAnalyBcs”byBisias,Flood,Lo,Valavanis,2012.
• Dynamic/mul2-periodrisk
Algorithms,GameTheory&Risk-averseDecisionMakingEvdokiaNikolova
• Expresspreferencesoverrandomvariables(loteries)bymappingeachtoasinglenumberthrougha“u2lityfunc2on”.[Bernoulli1738]
RiskI:ExpectedU2lityTheory
Algorithms,GameTheory&Risk-averseDecisionMakingEvdokiaNikolova
• Expresspreferencesoverrandomvariables(loteries)bymappingeachtoasinglenumberthrougha“u2lityfunc2on”.[Bernoulli1738]
• U2lityexistsassumingindependence,con2nuityaxioms
RiskI:ExpectedU2lityTheory
Algorithms,GameTheory&Risk-averseDecisionMakingEvdokiaNikolova
• Expresspreferencesoverrandomvariables(loteries)bymappingeachtoasinglenumberthrougha“u2lityfunc2on”.[Bernoulli1738]
• U2lityexistsassumingindependence,con2nuityaxioms• Experiment:
RiskI:ExpectedU2lityTheory
Algorithms,GameTheory&Risk-averseDecisionMakingEvdokiaNikolova
$48,000$55,000w.p.33%$48,000w.p.66%0w.p.1%
$55,000w.p.33%0w.p.67%
$48,000w.p.34%0w.p.66%
• Expresspreferencesoverrandomvariables(loteries)bymappingeachtoasinglenumberthrougha“u2lityfunc2on”.[Bernoulli1738]
• U2lityexistsassumingindependence,con2nuityaxioms• Experiment:
• Convexu2lity=>risk-lovingpreference• Concaveu2lity=>risk-aversepreference
RiskI:ExpectedU2lityTheory
Algorithms,GameTheory&Risk-averseDecisionMakingEvdokiaNikolova
$48,000$55,000w.p.33%$48,000w.p.66%0w.p.1%
$55,000w.p.33%0w.p.67%
$48,000w.p.34%0w.p.66%
RiskII:Mean-varianceframework• Ininvestmentscience,Markowitz(1952)iden2fiedriskwith
thevarianceofaporiolio• E.g.rou2ngunderuncertainty:
minimize(mean+standarddevia2on)oftravel2meRelated:
minimizevariances.t.meanisatmostathresholdminimizetripbudgets.t.Prob(late)<5%[valueatrisk]
• Cri2cisms
– Non-monotonicity(maypreferstochas2callydominatedsolu2ons)– Non-convexity(conflictsrisk-diversifica2on?)
Algorithms,GameTheory&Risk-averse
DecisionMakingEvdokiaNikolova
RiskIII:Coherentriskmeasures• Axioma2capproachtotheconstruc2onofriskmeasures
(Artzneretal.1999)– A1[Monotonicity]:X≥0impliesR(X)≤0– A2[Convexity]:R(βX+(1-β)Y)≤βR(X)+(1-β)R(Y)– A3[PosiBvehomogeneity]:R(βX)=βR(X)– A4[TranslaBoninvariance]:R(X+c)=R(X)–cforallrealc
• Example:Condi2onalValueatRisk,CVaRα(X):thecondi2onalexpecta2onoflossesthatexceedVaRα(X);Xiscon2nuous.
• Remark:whenXisnormallydistributed,bothVaRα(X)and
CVaRα(X)areequaltoE[X]–kStdev(X)forappropriateconstantsk.
Algorithms,GameTheory&Risk-averse
DecisionMakingEvdokiaNikolova
Implica2onsofriska{tudes
Algorithms,GameTheory&Risk-averseDecisionMaking
• Uncertaintyandriskcanaffectourdesignandanalysisof:
• Algorithms• AlgorithmicGameTheory• AlgorithmicMechanismDesign
EvdokiaNikolova
EvdokiaNikolova Algorithms,GameTheory&Risk-averseDecisionMaking
Algorithmicchallenges• E.g.,Inshortestpaths,op*malsubstructure(addi*vity)
propertyallowsforefficientdynamicprogrammingalgorithms(Dijkstra,Bellman-Ford,etc)
AB C
5
6.9
5
• Nonaddi2vity:inrisk-averseshortestpaths,op*malsubstructurefails.
• Non-convex(concave)objec2veEvdokiaNikolova Algorithms,GameTheory&Risk-averse
DecisionMaking
Algorithmicchallenges
AB C
Mean5St.dev.3
Mean6.9St.dev.1
Mean5St.dev.1
AB-top:mean+st.dev.=8AB-botom:mean+st.dev.=7.9
AC-top:mean+st.dev.=10+√10≈13.2AC-botom:mean+st.dev.=11.9+√2≈13.3
Op2malpath
Algorithms,GameTheory&Risk-averseDecisionMaking
AlgorithmicinsightsInsight1:Concaveobjec2ve
Insight2:Visualizeonmean-
varianceplaneInsight3:Solu2onisan
extremepointonmean-variancefron2er
minpathmean+r√pathvar{paths}
mean
varMean-VarianceofPaths
c
Exactalgorithmforrisk-averseshortestpathhasrun2men^O(logn).[Nikolova,Kelner,Brand,Mitzenmacher2006]OPEN:Isrisk-averseshortestpathinP?Onplanargraphs?
EvdokiaNikolova
Algorithms,GameTheory&Risk-averseDecisionMaking
AlgorithmicinsightsInsight1:Concaveobjec2ve
Insight2:Projectonmean-
varianceplaneInsight3:Solu2onisan
extremepoint
minrisk-aversefunc2on{feas.set}
mean
varMean-VarianceofPaths
c
There is an (1+ε)-approximation algorithm for risk-averse problem that uses poly( |input|, 1/ε) queries to exact algorithm for corresponding deterministic problem. [Nikolova2010]
EvdokiaNikolova
Algorithmicchallenges
Algorithms,GameTheory&Risk-averseDecisionMaking
• Challengeisnonlinearobjec2vefunc2onovercombinatorialfeasibleset
• Examplefromelectricitygrid
EvdokiaNikolova
Resistance
Ac2vepower
Reac2vepower
SuccessorsofedgeeinspanningtreeST
Implica2onsofriska{tudes
Algorithms,GameTheory&Risk-averseDecisionMaking
• Uncertaintyandriskcanaffectourdesignandanalysisof:
• Algorithms• AlgorithmicGameTheory• AlgorithmicMechanismDesign
EvdokiaNikolova
Algorithms,GameTheory&Risk-averseDecisionMaking
Risksensi2vityofpriceofanarchy• [Piliouras,Nikolova,Shamma2013]considerrou2nggameswith
uncertaindelaysresul2ngfrom“uniformschedulers”• Priceofanarchyoflinearconges2ongamesunderriska{tudes:
– Wald’sminimaxcost 2– Savage’sminimaxregret [4/3,1]– MinimizingExpectedcost 5/3– Averagecaseanalysis 5/3– Win-or-Go-Home unbounded – Secondmomentmethod unbounded
• Conclusion:RiskcriBcallyaffectspredicBonsofsystemperformance
• Relatedworkonrisk-aversioninrou2nggames:Ordóñez&S2er-Moses’10,Boyles-Kockelman-Waller’10(tolls),Nikolova&S2er-Moses’11-’14,‘15,Nie’11,Angelidakis-Fotakis-Lianeas’13,Comine{-Torico’13,Meir-Parkes’15,Lianeas-Nikolova-S2er-Moses’16,etc.
EvdokiaNikolova
EffectofRiskvsSelfishBehavior?
• CostoftrafficpaternC(x):centralplannerisrisk-neutral(althoughusersarerisk-averse),soweconsiderthesumofexpectedtravel*mes
• PriceofRiskAversion:capturesinefficiencyintroducedbyuserrisk-aversionw.r.t.torisk-neutralusers
risk-averseequilibriumrisk-neutralequilibrium
)C(x )C(xsup 0
instancesproblem
r
Algorithms,GameTheory&Risk-averseDecisionMakingEvdokiaNikolova
• Ingraphswithgeneralmean,variancefunc2onswhereusersminimize(mean+r*variance):
Cost(Risk-averseeq.)≤(1+ηrk)Cost(Risk-neutraleq.)• η=1forseries-parallelgraphs,η=2forBraessgraph,
η≤|V|/2forageneralgraph
• [Lianeas,Nikolova,S2er-Moses’16,*]• Aboveboundis2ghtforgeneralgraphs.• Boundcanalsobefoundw.r.tlatencyfunc2onclasses:• Priceofriskaversion≤(1+rk)PoA
Risk-averseselfishrou2ng*
Algorithms,GameTheory&Risk-averseDecisionMakingEvdokiaNikolova
*Lianeas,Nikolova,S2erMoses.“Risk-averseselfishrouBng.”ForthcominginMathema2csofOpera2onsResearch.
• Ingraphswithgeneralmean,variancefunc2onswhereusersminimize(mean+r*variance):
Cost(Risk-averseeq.)≤(1+ηrk)Cost(Risk-neutraleq.)• η=1forseries-parallelgraphs,η=2forBraessgraph,
η≤|V|/2forageneralgraph
• Aboveboundis2ghtforgeneralgraphs.• Boundcanalsobefoundw.r.tlatencyfunc2onclasses:
Priceofriskaversion≤(1+rk)PoA
Risk-averseselfishrou2ng*
Algorithms,GameTheory&Risk-averseDecisionMakingEvdokiaNikolova
*Lianeas,Nikolova,S2erMoses.“Risk-averseselfishrouBng.”ForthcominginMathema2csofOpera2onsResearch.
Implica2onsofriska{tudes
Algorithms,GameTheory&Risk-averseDecisionMaking
• Uncertaintyandriskcanaffectourdesignandanalysisof:
• Algorithms• AlgorithmicGameTheory• AlgorithmicMechanismDesign
EvdokiaNikolova
Effectofriskonmechanismdesign• Peoplerespondbetertoloteries(examplesin
transporta2on,energy,recycling,etc.)• Whataretheop2malloteries?I.e.whatistheop2mal
mechanismdesignifpeoplehavegivenrisk-averseorrisk-lovinga{tudes?
• Workinprogressonrisk-lovingmechanismdesignwithManolisPountourakis,GerYangatUTAus2n.
• Wanttoknowmore?TalktoManolis(SimonsFellow)Algorithms,GameTheory&Risk-averse
DecisionMakingEvdokiaNikolova
$48,000$55,000w.p.33%$48,000w.p.66%0w.p.1%
$55,000w.p.33%0w.p.67%
$48,000w.p.34%0w.p.66%
Conclusion
• CStheoristsliketoprovetheorems(andhaveguaranteesonalgorithmrun2mes,etc)
• CStheoristslike“cleanformula2ons”thatabstractawaymanyengineeringdetailssoastofindandunderstandcorealgorithmicchallenges(reducetotoypuzzles)
• Openques2ons?– Dynamic/Adap2vemodels?(streamingalgorithms,learningalgorithmsforreal-2medecisionmaking)
Algorithms,GameTheory&Risk-averseDecisionMakingEvdokiaNikolova