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CS 580: Algorithmic Game Theory

创建日期：2024-06-08 15:52:52

所属分类：计算机computer science

标签： CS 580

Algorithmic Game Theory

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CS 580: Algorithmic Game Theory

Instructions:
1. Please read all the instructions and questions carefully before you start answering the ques-
tions. Also, please ask for clarifications early on (on Slack or Piazza).
2. This exam has four questions, 25 points each. That is worth 100 points. However, it will be
graded out of 75 points to allow you to drop some questions.
3. You can work with at most one other person on this exam, but you need to write the solution
on your own and submit. Please mention the name of the student you discussed with on your
submission. Completely identical solutions will be considered plagiarism and a violation of
the university code of conduct.
4. Please type your solutions, if possible, in Latex or doc, whichever is suitable, and submit
them on Gradescope.
5. Even if you cannot solve a problem completely, submit whatever you have. Partial proofs,
high-level ideas, examples, and so on.
6. Except where otherwise noted, you may refer to lecture slides/notes. You cannot refer to
textbooks, handouts, or research papers that have not been listed. If you do use any approved
sources, make sure you cite them appropriately, and make sure to write in your own
words.
7. No late submissions will be accepted.
1. Potential and Smoothed games
(a) (10 points) This problem develops some theory about potential games; we talked about
these while discussing selfish routing. We consider an abstract finite game with n players,
each with finite strategy sets, say Si for agent i ≤ n. Each player has a payoff function
πi mapping outcomes (elements of S1×· · ·×Sn) to real numbers. Recall that a potential
function for such a game is defined by the following property: for every outcome s ∈
S1 × · · · × Sn, every player i, and every deviation s′i ∈ Si.
πi(s
′
i, s−i)− πi(si, s−i) = Φ(s′i, s−i)− Φ(si, s−i).
A team game is a game in which all players have the same payoff function: π1(s) = · · · =
πn(s) for every outcome s. In a dummy game, the payoff of every player i is independent
of its strategy: πi(si, s−i) = πi(s′i, s−i) for every s−i and every si, s
′
i ∈ Si.
1
Prove that a game with payoffs π1, . . . , πn is a potential game (i.e., admits a potential
function) if and only if it is the sum of a team game πt1, . . . , π
t
n and a dummy game
πd1 , . . . , π
d
n (i.e., πi(s) = π
t
i(s) + π
d
i (s) for every i and s.)
(b) (15 points) Consider n identical machines and m selfish jobs (the players). Each job j
has a processing time pj . Once jobs have chosen machines, the jobs on each machine are
processed serially from shortest to longest. (You can assume that the pj ’s are distinct.)
For example, if jobs with processing times 1, 3, and 5 are scheduled on a common ma-
chine, then they will complete at times 1, 4, and 9, respectively. The following questions
concern the game in which players choose machines in order to minimize their comple-
tion times. The objective function as a planner is to minimize the total completion time∑m
j=1Cj , where Cj is the completion time job j.
i. (4 points) Define the rank Rj of job j in a schedule as the number of jobs on j’s
machine with processing time at least pj (including j itself). For example, if jobs
with processing times 1, 3, and 5 are scheduled on a common machine, then they
have ranks 3, 2, and 1, respectively.
Prove that in these scheduling games, the objective function value of an outcome
can also be written as
∑m
j=1 pjRj .
ii. (4 points) Prove that the following algorithm produces an optimal outcome: (i) sort
the jobs from largest to smallest; (ii) for i = 1, 2, . . . ,m, assign the ith job in this
ordering to machine i mod n (where machine 0 means machine n).
iii. (7 points) Prove that for every such scheduling game, the expected objective function
value of every coarse correlated equilibrium is at most twice that of an optimal
outcome.
Hint: The (λ, µ)-smoothness condition (see notes provided) was required for all pairs
s∗, s of outcomes. Weaken the definition so that this condition only needs to hold for
some optimal outcome s∗ and all outcomes s. Observe that PoA of coarse correlated
equilibria remains at most λ1−µ assuming only this weaker condition (with the same
proof as before). Prove that this scheduling game satisfies this weaker condition for
λ = 2 and µ = 0.
2. Nash equilibrium and Cost Sharing Games (25 points)
(a) (10 points) Consider an n player game in which each player has 2 strategies. For this
problem, think of the strategies as ‘on’ or ‘off’. For example, the strategy can be either
to participate in an event or not. Further more, assume that the game is symmetric, in
that all players have the same payoff functions, and that the payoff for a player depends
only on the strategy of the player and the number of players playing strategy ‘on’. So the
game is defined by 2n values: uon(k) and uoff (k), which denote the payoff for playing
the ‘on’ and ‘off’ strategies, assuming that k of the other players chose to play ‘on’ for
k = 0, · · · , n− 1.
Give a polynomial time algorithm to find a correlated equilibrium for such a game. Note
that the input to this game consists of the 2n numbers above. As usual, polynomial
means polynomial in this input length. You may use the fact that linear programming
is solvable in polynomial time.
(b) (15 points) Let us extend the model of network cost-sharing games that we covered
in class by
2
allowing each edge e to have a cost γe(x) that depends on the number x of agents that
use it. The joint cost γe(x) is again split equally between the x users of the edge. Assume
that each function γe is concave, meaning that
γe(i+ 1)− γe(i) ≤ γe(i)− γe(i− 1)
for each i = 1, 2, . . . , k − 1.
Note that, at a strategy profile P = (P1, . . . , Pn), if f
P
e many agents are using edge
e ∈ E then the cost of each individual agent i is Ci(P ) =
∑
e∈P
γe(fPe )
fPe
, and the total
cost is C(P ) = ∑e∈E:fPe >0 γe(fPe ). The optimum outcome is P ∗ ∈ argminP C(P ). Let
Hk denote the k-th Harmonic number, i.e. Hk =
∑k
i=1
1
i .
i. (6 points) Prove that in every network cost-sharing game with k players, there exists
a pure Nash equilibrium with cost at most Hk times that of an optimal outcome.
ii. (9 points) Prove that in every network cost-sharing game with k players, every strong
Nash equilibrium has cost at most Hk times that of an optimal outcome.
3. Fair division (25 points)
(a) (10 points) Consider the problem of fairly allocating chores instead of goods. Let M
be the set of m chores that need to be allocated to a set N of n agents. Agent i
incurs pain/disutility dij from chore j, and has additive disutility over subsets, i.e.,
for set S ⊆ M , di(S) =
∑
j∈S dij . Note that, agents want to minimize their overall
disutility/pain. Accordingly, an allocation A = {A1, A2, . . . , An} of chores is said to be
envy-free up to any item (EFX) if and only if for every agents i, k
di(Ai \ {c}) ≤ di(Ak), for all chores c ∈ Ai.
i. (3 points) Suppose the instance is binary, i.e., for each i ∈ N, j ∈ M , dij ∈ {0, 1}.
Design a polynomial-time algorithm to find an EFX allocation.
ii. (3 points) Suppose the instance is identical, i.e., for each j ∈M , dij = dj > 0, ∀i ∈
N . Design a polynomial-time algorithm to find an EFX allocation.
iii. (4 points) Suppose the instance is identically ordered. That is, for all agents i,
di1 ≥ di2 ≥ · · · ≥ dim > 0. Show through an example that the following modification
of the envy-cycle algorithm fails: Consider chores from the most painful to least
painful (from 1 through m). Starting with an empty allocation, allocate the next
unallocated chore to a sink in the envy-graph, recompute the envy-edges, and remove
cycles.
(b) (15 points) In this fair allocation problem, we are given a graph G = (V,E) and a set
of n agents. Every item is associated with a unique edge in G, that is the items set
is M = E. The valuations of the agents for the items are identical and additive, that
is, valuation of any agent i ≤ n for a subset S ⊂ M is v(S) = ∑g∈S v(g). We need
to allocate items to agents in a fair manner, with the additional constraint that the set
of edges that corresponds to each bundle needs to be acyclic in G. In other words, an
allocation A = (A1, A2, . . . , An) is valid if and only if Ei = {e ∈ E | e ∈ Ai} forms a
forest in G for every 1 ≤ i ≤ n.
3
i. (2 points) Consider two spanning trees T1 and T2 of G. Prove that for every edge
e ∈ T1, there exists an edge e′ ∈ T2 such that T1 \ {e} ∪ {e′} is a spanning tree.
In general, a stronger theorem holds, which states that there exists a bijection ϕ
from the edges of T1 to the edges of T2 such that, for every edge e ∈ T1, we have
that T1 \ {e} ∪ {ϕ(e)} is a spanning tree. You can use this stronger fact in the
subproblems below, but you don’t need to prove it.
ii. (12 points) Show that, for this fair allocation setting, if a feasible allocation A is not
EF1, then there exists a feasible swap such that the valuation of Ah - the highest
valued bundle violating EF1 with respect to the least valued bundle - drops by at
least a multiplicative factor of (1− 1/m2), or its cardinality decreases by one, where
m is the number of items.
iii. (6 points) Using the above, design an algorithm to find an EF1 allocation in this
setting.
4. (Revenue and Welfare) (25 points)
(a) (10 points) Consider the following problem of designing a posted price mechanism, which
relates to prophet inequalities. There are n buyers for a single item and we want to sell
the item to one of them. Each buyer i arrives at our shop with probability pi, and they
don’t arrive at all with probability 1 − pi. If buyer i arrives, their value for the item is
vi. The buyers choose to arrive or not independently of each other, and we know that∑n
i=1 pi = 1 (on expectation, one buyer will arrive).
Consider the following mechanism: When buyer i arrives, if we haven’t sold the item
yet, we sell it to i with probability 1/2, else we pass with probability 1/2. Show that
the expected value of this mechanism is a 1/4-approximation to the expected maximum
value maxi pivi.
Hint: Recall the proof of the prophet inequality; notice that
E[ALG] =
n∑
i=1
Pr[Item is sold to buyer i] · vi.
Use this, along with Markov’s inequality, to upper bound the probability the item is not
sold at all. Markov’s inequality says that for a non-negative random variable X and
value a > 0, we have Pr[X ≥ a] ≤ E[X]a .
(b) (5 points) This problem considers a variation of the Bulow-Klemperer theorem. Consider
selling k ≥ 1 identical items (with at most one given to each bidder) to bidders with
valuations drawn iid from F , where F is a regular distribution (i.e., the corresponding
ϕ−1 is a monotonically increasing function). Prove that for every n ≥ k, the expected
revenue of the Vickrey auction (with no reserve) with n + k bidders is at least that of
the Myerson’s optimal auction for F with n bidders.
Hint: Myerson’s optimal auction will be Vickrey with reserve ϕ−1(0), i.e., discard bids
below ϕ−1(0), give the item to k highest bidders and charge them max{(k + 1)th highest
bid, ϕ−1(0)}
(c) (10 points) In a combinatorial auction there are n bidders and m items. Bidder i has a
monotone valuation vi(·) where vi(S) is their value for item set S (and vi(S∪T ) ≥ vi(S)
for all S, T ). A Walrasian Equilibrium is a price vector p⃗ such that:
4
• Each buyer i selects to purchase a set Bi ∈ argmaxS{vi(S)−
∑
j∈S pj}.
• The sets Bi are disjoint, and
⋃
iBi = [m].
i. (2 points) Find an example of 2 items and 2 bidders where there is no Walrasian
equilibrium.
ii. (8 points) Prove that a Walrasian equilibrium exists for v1, · · · , vn if and only if the
optimum of the LP relaxation below (called the configuration LP) is achieved at an
integral point (i.e. where each xi,S ∈ 0, 1).
max
∑
i
∑
S
vi(S) · xi,S
s.t.
∑
S
xi,S = 1 ∀i∑
S:j∈S
∑
i
xi,S ≤ 1 ∀j
xi,S ≥ 0 ∀i ∀S
Note that, here xi,S = 1 implies set S is assigned to agent i.
Hint: Use strong duality of linear programming.

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BCPM0008 ECN 620 QBUS6830 ECSE-415 ECON5102 MGMT20005 MAT 653 ECO 480 COMP 5/600 CSSE 5600 ME 571 7CCMMS61 CS 550 COMP2865 MATH3075 STATS 506 QTM 220 ISYS2120 FIT5106 STAT 153 MKTG3112 ECON2102 ELE2018 COMP3031 MSIN0010 SCC461 MATH237 CS246 CSC3065 CSE 101 ICTWEB411 COMP9313 ECOS 2025 FN620 MATH11111 MATH 3018 KE1068 FINA 6112 C++ MISM 6202 MSIN0105 CSC8631 MECO6935 COMP4121 ACCT2011 DPBS1150 EEL 3701C F19PL OPM 661 CSE 310 CS918 BIOL30001 ACCT3563 BANK3600 MET CS622 MATH367 ACCT5907 Continuing COMP5048 COMP61021 CS3334 GSOE9210 CSCI-570 COMP 5322 HW 2 ALY-6020 IEOR 4706 DATA 200 EBUS504 CGE12412 COMP4108 7SSMM800 ECMM171P MATE50002 CS225 RS 6003 BH2274 ELEC3111 MAT6669 ECON-UA 227 PHIL 1012 7SSMM603 TB028A P6 ECS708 CSC 427 CS6476 CSE 250A A1A2 ITS61504 EECS 376 model HW6 UV0010 ECON 6074 MG4F7 CMP-7030Y INFO3008 BFIP013 ECON 5068 MATH6002 CS3483 PSTAT 131 C31FM CSOR W4246 AAEC 5307 DATA ELEC362 COMSM0047 ECN6006 MA20224 BST351 MATH 323 MATH323 AA2021 COMP3811 COMP2013 IB9X60 INVP001 EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 MAS316/414 ACCT2522 MSIN0181 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