Gatti: A gentle introduction to game theory and crowding games. Outline. Outline. Nicola Gatti

Transcription

1 gentle introduction to game theory and crowding games Nicola Gatti Dipartimento di Elettronica e Informazione Politecnico di Milano Italy Outline Examples Game theory groundings Searching for a Nash equilibrium Congestion games Equilibrium inefficiency Crowding games Outline Examples Game theory groundings Searching for a Nash equilibrium Congestion games Equilibrium inefficiency Crowding games

2 Games game captures: a strategic situations between a number of agents each agent has a goal potentially in conflict with the goals of the other agents agents can cooperate or not, in any case they are selfish What are the practical scenarios in which we deal with games? Game scenarios () Game scenarios ()

3 Game scenarios (3) Game scenarios (4) Game scenarios (5)

4 Game scenarios (6) Game scenarios (7) Outline Examples Game theory groundings Searching for a Nash equilibrium Congestion games Equilibrium inefficiency Crowding games

5 Game theory It provides formal/mathematical tools to model game situations gents ctions Outcomes gents preferences over the outcomes Solution concepts Game model game is characterized by a pair mechanism: it specifies the rules of the game strategies: they specify how agents play in the game solution is defined as a strategy profile, specifying the strategy of each agent, which satisfies the conditions of some solution concept, e.g., Nash equilibrium Problems in game theory Solution computation Input: a game and a solution concept Output: a solution Questions: is a solution concept computable? how is it computable? is it approximable? Mechanism design Input: agents strategies and desired properties Output: a mechanism Questions: can we induce agents to play some desired strategies? with which properties?

6 Game mechanisms M =(N,, X, f, U) Game mechanisms M =(N,, X, f, U) set of agents, e.g., N = {, } Game mechanisms M =(N,, X, f, U) set of agents actions, e.g., = {, }, = {rock, paper, scissors}, =

7 Game mechanisms M =(N,, X, f, U) set of outcomes, e.g., X = {tie, win, win } Game mechanisms M =(N,, X, f, U) outcome function, e.g., f(a, a) =tie f(rock, paper) =win... Game mechanisms M =(N,, X, f, U) utility function, e.g., U = {U,U } U i (win i )= U i (tie) =0 U i (win i )=

8 Example () agent rock paper scissors rock tie win win agent paper scissors win tie win win win tie Example () agent rock paper scissors rock 0,0 -,,- agent paper scissors,- 0,0 -, -,,- 0,0 Example () gents: a man and a woman ctions: attending a concert of either ach or Outcomes: both agents attend the same concert together or each agent attends a different concert Preferences: each agent prefers to attend a concert with the other rather than alone, the man prefers ach, the woman prefers

9 Example () woman ach ach together at ach man at ach, woman at man man at, woman at ach together at Example () woman ach ach, 0,0 man 0,0, Uncertainty () The man is not certain about the intentions of the woman: she could be interested in meeting the man or she could prefer not to meet him How the situation can be modeled? ayesian games

10 Uncertain () The woman can be of two types differing in their preferences type : she desires to meet the man type : she desires not to meet the man Each type can occur with a given probability type :! type :! The woman perfectly knows the preferences of the man Example () woman type woman type ach ach man ach together at ach man at, woman at ach man at ach, woman at together at ach together at ach man at, woman at ach!! man at ach, woman at together at Example () woman type woman type ach ach ach, 0,0 ach,0 0, man 0,0, 0,,0!!

11 Strategies strategy defines the behavior of an agent pure: it prescribes the agent to play an action mixed: it prescribes the agent to randomize over a number of actions Pure strategies are degenerate mixed strategies Strategies can be represented as vectors of probabilities over all the actions of an agent Example () agent rock paper scissors agent rock paper scissors 0,0 -,,-,- 0,0 -, -,,- 0, Example () agent rock paper scissors agent rock paper scissors 0,0 -,,-,- 0,0 -, -,,- 0,

12 Example () woman ach man ach, 0,0 0,0, Example () woman ach man ach, 0,0 0,0, Solution concept solution concept defines a set of constraints over the strategies of the agents Why? Differently from single-objective optimization, it is not clear how to define what is the global optimum Differently from multi-objective optimization, agents are in conflict and therefore a generic Pareto efficient solution may be not appropriate We could have different situations in terms of information the agents have about the opponents

13 Example () agent Confess Defeat agent Confess Defeat 3,3 0,4 4,0, Example () agent Confess Defeat agent Confess Defeat 3,3 0,4 4,0, Nash equilibrium It is the most famous solution concept It captures situations in which No agent can communicate with the others before acting The information is complete and common It constraints the strategy of each agent to the best given the strategies of the other agents No agent can improve its utility by deviating unilaterally from a Nash equilibrium Nash equilibrium is not required to be resilient to coalition deviations

14 Example () woman ach ach, 0,0 man 0,0, Example () woman ach ach, 0,0 man 0,0, Example () woman ach ach, 0,0 man 0,0,

15 Example () agent Confess Defeat agent Confess Defeat 3,3 0,4 4,0, Example () agent Confess Defeat agent Confess Defeat 3,3 0,4 4,0, Outline Examples Game theory groundings Searching for a Nash equilibrium Congestion games Equilibrium inefficiency Crowding games

16 Searching for a pure Nash C D E F 0,,4 5,5 -,7-4,- 3,9 9,9 5, 0,0,0,0 4,5 C 7,7 0,0 6,8 3, 4,- 5,0 D 5,3 0, 5, 9,9 7, 0,0 E 3,0 3,3 0, 5, 0,4 0,4 F,6 5,5 3, 6,0 -,-3 8,3 Searching for a pure Nash C D E F 0,,4 5,5 -,7-4,- 3,9 9,9 5, 0,0,0,0 4,5 Generate randomly a starting point C 7,7 0,0 6,8 3, 4,- 5,0 D 5,3 0, 5, 9,9 7, 0,0 E 3,0 3,3 0, 5, 0,4 0,4 F,6 5,5 3, 6,0 -,-3 8,3 Searching for a pure Nash C D E F 0,,4 5,5 -,7-4,- 3,9 9,9 5, 0,0,0,0 4,5 C 7,7 0,0 6,8 3, 4,- 5,0 D 5,3 0, 5, 9,9 7, 0,0 E 3,0 3,3 0, 5, 0,4 0,4 F,6 5,5 3, 6,0 -,-3 8,3

17 Searching for a pure Nash C D E F C D 0,,4 5,5 -,7-4,- 3,9 9,9 5, 0,0,0,0 4,5 7,7 0,0 6,8 3, 4,- 5,0 5,3 0, 5, 9,9 7, 0,0 Check whether it is a Nash: If yes, conclude If not, put it in a Tabu list E 3,0 3,3 0, 5, 0,4 0,4 F,6 5,5 3, 6,0 -,-3 8,3 Searching for a pure Nash C D E F 0,,4 5,5 -,7-4,- 3,9 9,9 5, 0,0,0,0 4,5 C 7,7 0,0 6,8 3, 4,- 5,0 D 5,3 0, 5, 9,9 7, 0,0 E 3,0 3,3 0, 5, 0,4 0,4 F,6 5,5 3, 6,0 -,-3 8,3 Searching for a pure Nash C D E F C 0,,4 5,5 -,7-4,- 3,9 9,9 5, 0,0,0,0 4,5 7,7 0,0 6,8 3, 4,- 5,0 Select an agent and move on the best response for such an agent excluded those in the Tabu list D 5,3 0, 5, 9,9 7, 0,0 E 3,0 3,3 0, 5, 0,4 0,4 F,6 5,5 3, 6,0 -,-3 8,3

18 Searching for a pure Nash C D E F 0,,4 5,5 -,7-4,- 3,9 9,9 5, 0,0,0,0 4,5 C 7,7 0,0 6,8 3, 4,- 5,0 D 5,3 0, 5, 9,9 7, 0,0 E 3,0 3,3 0, 5, 0,4 0,4 F,6 5,5 3, 6,0 -,-3 8,3 Searching for a pure Nash C D E F C D 0,,4 5,5 -,7-4,- 3,9 9,9 5, 0,0,0,0 4,5 7,7 0,0 6,8 3, 4,- 5,0 5,3 0, 5, 9,9 7, 0,0 Check whether it is a Nash: If yes, conclude If not, put it in a Tabu list E 3,0 3,3 0, 5, 0,4 0,4 F,6 5,5 3, 6,0 -,-3 8,3 Searching for a pure Nash C D E F 0,,4 5,5 -,7-4,- 3,9 9,9 5, 0,0,0,0 4,5 C 7,7 0,0 6,8 3, 4,- 5,0 D 5,3 0, 5, 9,9 7, 0,0 E 3,0 3,3 0, 5, 0,4 0,4 F,6 5,5 3, 6,0 -,-3 8,3

19 Searching for a pure Nash C D E F C 0,,4 5,5 -,7-4,- 3,9 9,9 5, 0,0,0,0 4,5 7,7 0,0 6,8 3, 4,- 5,0 Select an agent and move on the best response for such an agent excluded those in the Tabu list D 5,3 0, 5, 9,9 7, 0,0 E 3,0 3,3 0, 5, 0,4 0,4 F,6 5,5 3, 6,0 -,-3 8,3 Searching for a pure Nash C D E F 0,,4 5,5 -,7-4,- 3,9 9,9 5, 0,0,0,0 4,5 C 7,7 0,0 6,8 3, 4,- 5,0 D 5,3 0, 5, 9,9 7, 0,0 E 3,0 3,3 0, 5, 0,4 0,4 F,6 5,5 3, 6,0 -,-3 8,3 Searching for a pure Nash

20 Searching for a pure Nash C D E F 0,,4 5,5 -,7-4,- 3,9 9,9 5, 0,0,0,0 4,5 C 7,7 0,0 6,8 3, 4,- 5,0 D 5,3 0, 5, 9,9 7, 0,0 E 3,0 3,3 0, 5, 0,4 0,4 F,6 5,5 3, 6,0 -,-3 8,3 Searching for a pure Nash C D E F C 0,,4 5,5 -,7-4,- 3,9 9,9 5, 0,0,0,0 4,5 7,7 0,0 6,8 3, 4,- 5,0 Select an agent and move on the best response for such an agent excluded those in the Tabu list D 5,3 0, 5, 9,9 7, 0,0 E 3,0 3,3 0, 5, 0,4 0,4 F,6 5,5 3, 6,0 -,-3 8,3 Searching for a pure Nash

21 Searching for a pure Nash C D E F 0,,4 5,5 -,7-4,- 3,9 9,9 5, 0,0,0,0 4,5 C 7,7 0,0 6,8 3, 4,- 5,0 D 5,3 0, 5, 9,9 7, 0,0 E 3,0 3,3 0, 5, 0,4 0,4 F,6 5,5 3, 6,0 -,-3 8,3 Searching for a pure Nash C D E F C D 0,,4 5,5 -,7-4,- 3,9 9,9 5, 0,0,0,0 4,5 7,7 0,0 6,8 3, 4,- 5,0 5,3 0, 5, 9,9 7, 0,0 est response paths There can be multiple best response paths even for the same equilibrium E 3,0 3,3 0, 5, 0,4 0,4 F,6 5,5 3, 6,0 -,-3 8,3 Searching for a pure Nash C D E F C D 0,,4 5,5 -,7-4,- 3,9 9,9 5, 0,0,0,0 4,5 7,7 0,0 6,8 3, 4,- 5,0 5,3 0, 5, 9,9 7, 0,0 This is not a best response path Starting from (,), the best response path would cycle E 3,0 3,3 0, 5, 0,4 0,4 F,6 5,5 3, 6,0 -,-3 8,3

22 Searching for a pure Nash C D E F 0,,4 5,5 -,7-4,- 3,9 9,9 5, 0,0,0,0 4,5 C 7,7 0,0 6,8 3, 4,- 5,0 D 5,3 0, 5, 9,9 7, 0,0 E 3,0 3,3 0, 5, 0,4 0,4 F,6 5,5 3, 6,0 -,-3 8,3 Pure Nash non-existence agent rock paper scissors rock 0,0 -,,- agent paper scissors,- 0,0 -, -,,- 0,0 Nash existence Every finite game admits at least a Nash equilibrium in mixed strategies [Nash, 950] The proof is by rouwer s fixed point theorem nyway, a game may not admit any pure strategy Nash equilibrium The Nash theorem does not provide any procedure to find mixed strategy Nash equilibria

23 Example agent rock paper scissors rock 0,0 -,,- 3 agent paper,- 0,0 -, 3 scissors -,,- 0, Nash and complexity Given a game with n agents, the decision problem is there a pure Nash? is in P Given a game with n agents, what is the computational complexity to find a mixed strategy? The decision problem is there a mixed Nash? has always answer YES Searching for a Nash equilibrium is in TFNP (Total Functional NP: search problems always admitting solutions) TFNP is a subclass of NP (the verification problem is in P) Known results NashSearch is not NP-complete unless NP = co-np NashSearch is PPD-complete PPD is an ad hoc class for fixed point problems It is commonly believed that PPD is not a subclass of P Why NashSearch is hard? The problem is combinatorial in the actions played with strictly positive probability at the equilibrium With n actions per agent, the number of possible different sets of actions over which each can randomize is O( n ) The number of possible joint sets of actions is if the number of agents is O(4 n )

24 Outline Examples Game theory groundings Searching for a Nash equilibrium Congestion games Equilibrium inefficiency Crowding games Congestion games The game is based on the congestion of some resources gent can choose one or more resources Each agent, choosing a resource, congests the resource (potentially with different weight) Each agent has a (potentially different) cost depending on the resource and on the congestion Example () Woman she needs to go to 3 she has two options: --3, -4-3 Man he needs to go to he has two options: 4--, c{3,4}()= c{3,4}()=5 c{,4}()=, c{,4}()=3 3 c{,3}()=4, c{,3}()=6 c{,}()= c{,}()=4

25 Example () woman ,-5-6,-8 man ,-7-8,-7 Example () woman ,-5-6,-8 man ,-7-8,-7 Resources: gents: Weights: Congestions: Costs: Utilities: u = The model R = {,...,r} N = {,...,n} W = {w,...,w n } l i = P jn,j!i w j L = {l,...,l r } C = {c i,j (l j ):i N,j R} c

26 Properties Single-choice congestion games Every user can choose a single resource Weighted congestion games Different agents have different weights Player-specific congestion games Different agents have different costs Notable classes General congestion games Unweighted, non-player-specific congestions games Crowding games Single-choice, unweighted, player-specific congestion games with costs non-decreasing in the congestion General congestion games Theorem [Rosenthal, 973]: every general congestion game has at least one pure Nash equilibrium Computing a Nash equilibrium in general congestion games is easy How can we compute it in efficient way?

27 Exact potential game game in which there a potential function such that: (a i,a i ) (a 0 i,a i)=u(a i,a i ) u(a 0 i,a i) agent agent rock paper rock paper agent rock paper b+w,c+w -b-w,-c-w b-w,c-w -b+w,-c+w agent rock paper b+c+w -b+c-w b-c-w -b-c+w Example agent agent rock paper rock paper agent rock paper, 9,0 0,9 6,6 agent rock paper???? Example agent agent rock paper rock paper agent rock paper, 9, ,6 agent rock paper

28 Potential games and pure Nash Consider a pure strategy profile that: (a,...,a n ) (a i,a i ) (a 0 i,a i) 0, 8a 0 i,i such y definition of potential function we have: (a i,a i ) (a 0 i,a i) 0! u(a i,a i ) u(a 0 i,a i) 0, 8a 0 i,i This entails that (a,...,a n ) is a Nash equilibrium Corollary: every potential game admits a pure Nash equilibrium Potential function and NashSearch Every local maximum of the potential function is a Nash equilibrium We can find a Nash equilibrium by searching for a local maximum local maximum can be achieved by local optimization, e.g., iterative improvement The problem is PLS-complete Other potential games Classes Ordinal potential function Weighted potential function Generalized ordinal potential function They are relaxations of the definition of the exact potential function lso with these functions: local maximum corresponds to a pure Nash There is a pure Nash

29 Potential and congestion games Every congestion game is a potential game Every congestion game admits a potential function Every congestion game has a pure Nash Every potential game is isomorphic to a congestion game Finite improvement property (FIP) n improvement path is a sequence of strategy profiles in which two consecutive profiles differ for: The strategy of only one agent (said deviator) changes The deviator improves its utility game satisfies the FIP if every improvement path is finite Then, every improvement path leads to a Nash equilibrium (otherwise the path would be infinite) congestion game satisfies FIP est response vs improvement paths est response paths are a subset of improvement paths est response paths require that the deviator moves on the best response Improvement paths require that the deviator moves on a strictly better strategy If a game satisfies FIP, then all the improvement paths are finite and therefore all the best response paths are finite The reverse is not true

30 Example C D E F 0,,4 5,5 -,7-4,- 3,9 9,9 5, 0,0,0,0 4,5 C 7,7 0,0 6,8 3, 4,- 5,0 D 5,3 0, 5, 9,9 7, 0,0 E 3,0 3,3 0, 5, 0,4 0,4 F,5 5,6 3, 6,0 -,-3 8,3 Example C D E F 0,,4 5,5 -,7-4,- 3,9 9,9 5, 0,0,0,0 4,5 C 7,7 0,0 6,8 3, 4,- 5,0 D 5,3 0, 5, 9,9 7, 0,0 E 3,0 3,3 0, 5, 0,4 0,4 F,5 5,6 3, 6,0 -,-3 8,3 Example C D E F C 0,,4 5,5 -,7-4,- 3,9 9,9 5, 0,0,0,0 4,5 7,7 0,0 6,8 3, 4,- 5,0 The game admits Finite est Response property D 5,3 0, 5, 9,9 7, 0,0 E 3,0 3,3 0, 5, 0,4 0,4 F,5 5,6 3, 6,0 -,-3 8,3

31 Example C D E F 0,,4 5,5 -,7-4,- 3,9 9,9 5, 0,0,0,0 4,5 C 7,7 0,0 6,8 3, 4,- 5,0 D 5,3 0, 5, 9,9 7, 0,0 E 3,0 3,3 0, 5, 0,4 0,4 F,5 5,6 3, 6,0 -,-3 8,3 Example C D E F C 0,,4 5,5 -,7-4,- 3,9 9,9 5, 0,0,0,0 4,5 7,7 0,0 6,8 3, 4,- 5,0 The game does not admit Finite Improvement property D 5,3 0, 5, 9,9 7, 0,0 E 3,0 3,3 0, 5, 0,4 0,4 F,5 5,6 3, 6,0 -,-3 8,3 Simultaneous deviations and FIP agent agent 0,0,, 0,0

32 Simultaneous deviations and FIP agent agent 0,0,, 0,0 Simultaneous deviations and FIP agent agent 0,0,, 0,0 Observations Why FIP is important? Given a number a set of agents, if the game has FIP any iterative improvements of the agents leads to a Nash equilibrium Why FR is important? Given a number a set of agents, if the game has FIP any best response dynamics leads to a Nash equilibrium In practice? gents can spontaneously achieve a Nash equilibrium without any sort of coordination No assumption is requires except that the agents can observe the current cost of the resources

33 Example Example Example cost = 0 cost = 5 cost = 4

34 Example cost = 0 cost = 5 cost = 4 Example cost = cost = 5 cost = 3 Example cost = cost = 5 cost = 3

35 Example cost = cost = 4 cost = 3 Example cost = cost = 4 cost = 3 Example cost = cost = 4 cost = 3

36 Example cost = cost = 4 cost = 3 Example cost = 3 cost = 3 cost = 3 Example cost = 3 cost = 3 cost = 3 Starting from any possible initial configuration, best response dynamics lead to a Nash equilibrium

37 Outline Examples Game theory groundings Searching for a Nash equilibrium Congestion games Equilibrium inefficiency Crowding games Nash equilibria and efficiency game could admit multiple Nash equilibria We do not know what will be the Nash equilibrium the agents will achieve Can we characterize them in some way? Example () cost = 3 cost = 3 cost = 3 Social cost = (= 3 x 9) = 7

38 Example () cost = 3 cost = 4 cost = Social cost = = 9 Example () cost = cost = cost = Social cost = + + = 3 Example () cost = cost = cost = 0 Social cost = + + = 5

39 Price of narchy (Po) It measures the inefficiency of equilibria Po = max xne P i c i(x) min x Pi c i(x) It considers the socially worst equilibrium and the best (non-equilibrium) allocation Obviously, Po If it is equal to, then the worst equilibrium is also an optimal allocation and, if there are more equilibria, they all give the same social cost Price of Stability (PoS) It measures the inefficiency of the equilibria PoS = min xne P i c i(x) min x Pi c i(x) It considers the socially best equilibrium and the best (non-equilibrium) allocation Obviously, PoS If it is, it means that the optimal allocation is an equilibrium If it is strictly larger than, it means that the optimal allocation cannot be achieved by RD Example +e / /3 /4 /k S S... S3 S4 Sk

40 Example Optimal allocation has a cost of: +e +e / /3 /4 /k S S... S3 S4 Sk Example Unique equilibrium has a cost of: +/ /k +e / /3 /4 /k S S... S3 S4 Sk Example +e / /3 /4 /k S S... S3 S4 Sk PoS = O(ln(k))

41 Equilibrium characterization? Worst allocation Po Worst Nash best response dynamics PoS est Nash inefficiency est allocation Equilibrium characterization? Worst allocation Po Worst Nash best response dynamics PoS est Nash some equilibrium could be never achieved by best response dynamics inefficiency est allocation Equilibrium characterization? Worst allocation Po Worst Nash best response dynamics PoS est Nash from the same starting point, different dynamics could lead to different equilibria inefficiency est allocation

42 Equilibrium characterization? Worst allocation Po Worst Nash best response dynamics PoS est Nash what is the length of the dynamics? inefficiency est allocation Game design Information over the equilibria can be used to design games For instance, designing games in which equilibria are as efficient as it is possible raess paradox c = n/00 c = 45 c = 45 c = n/00

43 raess paradox c = n/ c = c = c = n/00 raess paradox individual cost = /00 = 65 social cost = 65 x 4000 c = n/ c = c = c = n/00 raess paradox c = n/ c = c = 0 c = c = n/00

44 raess paradox c = n/ c = c = 0 c = c = n/00 raess paradox individual cost = 4000/ /00 = 80 social cost = 80 x 4000 c = n/ c = c = 0 c = c = n/00 raess paradox individual cost = 4000/ /00 = 80 social cost = 80 x 4000 c = n/ c = c = 0 c = c = n/00 introducing a new resource with zero cost makes the equilibrium less efficient!

45 Is raess paradox rare in practice? In Seoul, South Korea, a speeding-up in traffic around the city was seen when a motorway was removed as part of the Cheonggyecheon restoration project (008) In Stuttgart, Germany after investments into the road network in 969, the traffic situation did not improve until a section of newly-built road was closed for traffic again In 990 the closing of 4nd street in New York City reduced the amount of congestion in the area In 008 Youn, Gastner and Jeong demonstrated specific routes in oston, New York City and London where this might actually occur and pointed out roads that could be closed to reduce predicted travel times Outline Examples Game theory groundings Searching for a Nash equilibrium Congestion games Equilibrium inefficiency Crowding games Crowding and FIP Crowding games with 3 ore more agents may not admit FIP Every unweighted crowding game admits a pure Nash equilibrium Weighted crowding games may not admit pure Nash There always exists a best response path with a length L apple r n+ ut, there could exist cycles and, thus, best response path could be infinite

46 Example l = 0 l = 5 l = 4 Example l = 0 l = 4 l = 5 Example l = 4 c=4 c= c=4 l = 0 c=4 c=5 c=5 c=5 l = 5 c=.5 c=.5

47 Example l = 3 c= c=.5 c=3 l = c=3 c=5 c=5 c=5 l = 5 c=.5 c=.5 Example l = 3 c= c=.5 c=3 l = c= c=4 c=3 c=4 l = 4 c= c= Example l = 3 c=3 c=.5 c=3 l = 3 c=3 c=3 c=.5 c=.5 l = 3 c=.5 c=.5

48 Example l = 3 c=4 c=.5 c=3 l = 4 c= c=3 c=3 c= l = c= c= Weighted crowding game l = 0 l = 5 w= w= w= w= w= w= w= l = 7 w= w= Crowding games are sequentially solvable Sequential version of a strategic-form game Woman she needs to go to 3 she has two options: --3, -4-3 Man he needs to go to he has two options: 4--, c{3,4}()= c{3,4}()=5 c{,4}()=, c{,4}()=3 3 c{,3}()=4, c{,3}()=6 c{,}()= c{,}()=4

49 Crowding games are sequentially solvable Sequential version of a strategic-form game woman man ,-5-6,-8-9,-7-8,-7 Crowding games are sequentially solvable Sequential version of a strategic-form game woman man man ,-5-6,-8-9,-7-8,-7 woman woman -4,-5-6,-8-9,-7-8,-7 Crowding games are sequentially solvable Sequential version of a strategic-form game woman man man ,-5-6,-8-9,-7-8,-7 woman woman -4,-5-6,-8-9,-7-8,-7

50 Crowding games are sequentially solvable Sequential version of a strategic-form game woman man man ,-5-6,-8-9,-7-8,-7 woman woman -4,-5-6,-8-9,-7-8,-7 Sequentially solvable games Definition Take a game Call the set of its Nash equilibria NE Consider a sequential version of the game Call the set of its subgame perfect equilibria SPE game is sequentially solvable if and only is SPE is a subset of NE for every sequential versions (i.e., for every order over the agents) Fixed an order over the agents, they can achieve a Nash equilibrium by playing the associated extensive-form game Negative example game may be not sequentially solvable agent H T agent agent H T,- -, -,,- agent agent,- -, -,,-

51 Crowding games Unweighted Crowding games are sequentially solvable Weighted crowding games may not admit pure Nash equilibria and therefore they may be not sequentially solvable Coalitional issues Nash equilibrium captures exclusively situations in which no agent can gain more by unilateral deviations What happens when agents can form coalitions? Strong Nash equilibrium (if all the possible coalitions are considered) Coalitional equilibrium (if a subset of all the possible coalitions are considered) These solution concepts may not exist even in mixed strategies