CSCI 3210: Computational Game Theory Linear Programming and - - PDF document
2/14/18 CSCI 3210: Computational Game Theory Linear Programming and 2-Player Zero-Sum Games Ref: Wikipedia and [AGT] Ch 1 Mohammad T . Irfan Email: mirfan@bowdoin.edu Web: www.bowdoin.edu/~mirfan Course Website:
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u Prove that NE exists– in two ways
u
Doesn't give an algorithm (why?)
u Gives an algorithm
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u Penalty kick game Left (0.42) Right (0.58) Left (0.38)
Right (0.62)
Shooter Goalkeeper
u Assumption (wlog): sum of payoffs in each
u More than 2 actions?
u Need an algorithm
L R U
D
Row player Column player L R U
D
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u Optimization
u Production, machine scheduling, employee
scheduling, supply chain management, etc.
u Game theory u In general: optimization
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u We can choose the values of these variables u What's the goal? u What values can we choose?
u Minimization or maximization u Must be linear in the variables
u Restricts the values of choice variables u Must be linear in the variables
u One is planning his day-to-day life. Outside of 10
hours of sleep every day, he wants to set aside a few hours for studying and a few hours for connecting with friends.
u Gets 10 units/hr of payoff from study and 20
units/hr of payoff from connecting with friends.
u Must study at least 6 hours every day. Also, feels
guilty if spends more than 6 hours with friends.
u How should he allocate time optimally?
u Variables? u Objective function? u Constraints?
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u Want to study at least 10 hours/day and do
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u Gets 15 units/hr of payoff for studying up to
u Wants at least 6 hours of study u Wants at most 6 hours of time with friends
u A tennis player is making a plan for
u She wants to practice service at least 100
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u Images from this tutorial:
u 4x1 matrix (AKA vector) u 3x3 matrix
u 2x3 matrix multiplied by 3x2 matrix u Result is a 2x2 matrix
must match
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u Transpose operator: superscript T u (A B)T = BT AT
A = 1 2 3 4 5 6 ! " # # # $ % & & & AT = 1 2 3 4 5 6 ! " # # $ % & &
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u Simplex (Dantzig, 1947)
u Worst case exponential time u Practically fast
u Ellipsoid (Khachiyan, 1979)
u O(n4 L) for n variables and L input bits u Pseudo-polynomial
u Karmarkar's algorithm (Karmarkar, 1984)
u O(n3.5 L) for n variables and L input bits u Pseudo-polynomial, but breakthrough for practical
reasons u Open problem: strongly polynomial algorithm?
u Interview with Dantzig
u http://www.personal.psu.edu/ecb5/Courses/
M475W/WeeklyReadings/Week%2015/ An_Interview_with_George_Dantzig.pdf u If the "primal" LP is maximization, its "dual"
u Every variable of the primal LP leads to a
u Dual of dual is primal.
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Source: Applied Mathematical Programming book
Primal Maximize cTx subject to: Ax <= b x >= 0 Dual Minimize bTy subject to: ATy >= c y >= 0
u How many Bowdoin logs and chocolate cakes should
Thorne make to maximize its revenue?
u Objective function. Each log has a satisfaction of 10 (or
price of $10), each cake 5.
u Constraints. For both desserts, the chef needs to use an
Processing time/log Processing time/cake Total available time Oven
5 min 1 min 85 min
Food processor
1 min 10 min 300 min
Boiler
4 min 6 min 120 min Derive primal and dual LP
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u Moulton wants to borrow Thorne's equipment
u Dual objective: minimize total cost of
u Dual constraints: recuperate lost payoff for
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u primal constraint non-binding (not equal) =>
u Similar condition holds for dual constr. & primal
var. u The reverse implication may not hold
Increasing
function
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u Any feasible solution of the dual LP (minimization) gives an
upper bound on the optimal solution of the primal LP (maximization).
u Proof
u Any feasible solution of the primal LP (maximization) is a
lower bound on the optimal solution of the dual LP (minimization). Primal LP (max) Dual LP (min)
Increasing
function Gap?
u Implications
u Primal unbounded è Dual infeasible u Dual unbounded è Primal infeasible
u Both primal and dual may be infeasible
Primal LP (max) Dual LP (min)
Increasing
function Gap?
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u If the primal LP has a finite optimal solution,
u In other words, if either the primal or the dual LP
has a finite optimal solution, the gap between them is 0.
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u Assumption (wlog): sum of payoffs in each
L R U
D
Row player Column player L R U
D
Matrix A
u How much (vr) can row guarantee from col.?
u Want largest vr possible
u Row: choose mixed strategy p (vector of
u Expected payoff of row when of col. plays j
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i
i
Row player's thought process: maximize my guaranteed payoff v from column player knowing that column player will always minimize her payout to me (in
action j of column player).
u How little (vc) can col. player pay to row? u Choose mixed strategy q (vector of
u Expected payoff of row player for action i =
u LP
minimize my guaranteed payout u to row player knowing that row player will choose to maximize her payoff.
Ai, jqj
j
≤ u, for each action i of row player qj =1
j
qj ≥ 0, for each action j of column player
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minimize my guaranteed payout u to row player knowing that row player will choose to maximize her payoff.
Ai, jqj
j
≤ u, for each action i of row player qj =1
j
qj ≥ 0, for each action j of column player
u At an equilibrium, vr = vc
u Proof:
+ bounded).
u Another proof:
payoff and v* cannot be lower than that.
row player more than vr. So, vr = v*. SImilarly, vc = v*.
u This quantity vc or vr is known as the value of
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u Given a graph, select the minimum number
u Answer?
4 5 6 7 8 9 2 3 1
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u xt is 1 if node t is picked and 0 otherwise u Is the following a usual linear program?
Integer linear program (ILP) In general "NP-hard" to solve
Approximation algorithm: Rounding the LP solution gives a 2-approximation
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u Select the maximum number of edges so that
u What's your solution?
4 5 6 7 8 9 2 3 1
u Primal
u Dual
c and b are vectors of 1s. Maximum matching problem (relaxed)