Game Theoretic Analysis in Supply Chains

1
Game Theoretic Analysis in Supply Chains

• Yang Sun
• Dept. of Industrial Engineering
• Arizona State University

2
Agenda

• Basic introduction to game theory
• Basic economic models (market games)
• Supply chain model Competitive newsvendors
• Capacity allocation game with non-competitive
  buyers
• Capacity allocation game with competitive buyers

3
Game Theory

• Definition Game theory is a branch of applied
  mathematics that studies interaction among a
  group of rational agents who behave
  strategically.
• Key Concepts
• Group
• Interaction
• Strategic
• Rational

4
A Brief Game Theory History

• Early works James Waldegraves, Antoine Augustin
  Cournot, Francis Ysidro Edgeworth, Emile Borel in
  1700s and 1800s
• 1928-1944 John von Neumann and Oskar Morgenstern
  published a series of papers and was credited as
  the fathers of modern game theory. Their work is
  culminated in the 1944 book The Theory of Games
  and Economic Behavior.
• 1950 Discussion and experimentation on the
  Prisoners dilemma at the RAND Corp.
• 1950 John Nashs dissertation on non-cooperative
  games
• Advisor Albert W. Tucker
• Contained the definition and properties of an
  optimum strategy that would later be called the
  Nash Equilibrium.
• Led to three important journal articles.
• 1950s Games of imperfect information (Kuhn
  1953) cooperative games (Aumann 1959) game
  theory experienced a flurry of activity including
  its first applications in philosophy and
  political science.
• 1960s Reinhard Seltens work on subgame perfect
  equilibria and John Harsanyis work on compete
  information and Bayesian games.
• 1970s and 1980s Works on evolutionary strategy
  and its applications in biology correlated
  equilibrium trembling hand equilibrium common
  knowledge sequential equilibrium extensive
  games repeated games extensive Bayesian games
  a lot of applications.
• 1994 Nash, Selten, and Harsanyi won the Nobel
  prize in Economics.
• 2005 Thomas Schelling (works on evolutionary
  game theory) and Robert Aumann (works on common
  knowledge) won the Nobel prize.

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Games and Strategies

• Static Game
• Dynamic Game
• Non-cooperative game
• Cooperative game
• Pure Strategies
• Mixed Strategies

• Two ways of formalizing a game
• The extensive form
• The normal form
• Players
• Strategies
• Payoffs
• (See examples)

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Examples
1
R
L
2
l
l
r
r
3 1
1 0
2 -1
2 0
Source Tirole, Jean, 1988, The Theory of
Industrial Organization, MIT Press
7
Examples (contd)
P2
P1
P2
P1
Source Tirole, Jean, 1988, The Theory of
Industrial Organization, MIT Press
8
The Prisoners Dilemma

• (D, D) is the dominant strategy equilibrium.
• Si is a dominant strategy if Ui(Si, S-i)
  gtUi(Si, S-i) for all Si, S-i
• i.e., the dominant strategy is independent of
  others strategies.
• A dominant equilibrium is a Nash equilibrium.

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Nash Equilibrium

• A strategy vector S(S1, S2,, Sn) is a Nash
  Equilibrium if for all i,
• Ui(Si, S-i) gtUi(Si, S-i) for all Si
• The concept was originally shown in the Cournot
  duopoly game (we will talk about this later).
• Nash showed for the first time in his
  dissertation, Non-cooperative games (1950), that
  Nash equilibria exist for finite games with any
  number of players. Until Nash, this had only been
  proved for 2-player zero-sum games by von Neumann
  and Morgenstern (1944)

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Other Important Concepts

• Sub-game perfect Nash equilibrium
• Complete Information and Common Knowledge
• Mixed Strategy equilibrium
• Bayesian equilibrium
• Note that not all games have an equilibrium.
• Many games have multiple equilibria.
• Some games have equilibria that are not good.

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An Economic Example

• Consider a monopoly who is the only one in the
  market that sells a particular product and there
  is no viable substitute goods.
• The price is set by market and is a function of
  the total sales quantity
• e.g. p D q, where D is a constant.
• The production cost is also a function of the
  quantity.
• e.g. C(q) cq, where c is a constant.
• The monopoly solves the following optimization
  problem to determine the optimal sales quantity
  that maximize his profit.
• The first-order condition yields

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The Market Game

• Assume two firms selling the same product. There
  are barriers to enter the market, so the two
  firms form a duopoly (oligopoly).
• The market price is a function of the total sales
  quantity.
• Payoffs (profits) to the two players are then

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Nash Equilibrium

• Nash Equilibrium is characterized by solving a
  system of best responses that translates into the
  system of first-order conditions
• This is the classic Cournot duopoly model.
• Are the firms more profitable?

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Oligopoly Models

• Collusion Model
• Stackelberg Model
• Cournot Model
• N-Firm Nash Cournot
• N? 8 Perfect Competition
• Recommended Readings
• Mas-Colell, Whinston, and Green, 2000,
  Microeconomic Theory
• Varian, 1992, Microeconomic Analysis
• Tirole, 1988, The Theory of Industrial
  Organization
• Osborne and Rubinstein, A Course in Game Theory

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A Supply Chain Example

• Consider the classic newsvendor model
• Each newsvendor buys Q units of a single product
  at the beginning of a single selling season.
• Demand during the season is a random variable D
  with distribution function FD and density
  function fD.
• Each unit is purchased for c and sold on the
  market for r gt c
• In the absence of competition, each newsvendor
  solves the following optimization problem
• with the unique solution

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Competitive Newsvendors

• Assume two newsvendors selling the same product.
  If retailer i is out of stock, all unsatisfied
  customers will try to buy at retailer j instead
  (competing on product availability).
• Retailer is total demand is Di (Dj Qj).
  Payoffs to the two players are then
• First-order conditions are
• See Cachon and Netessine (2004) and Lippman and
  McCardle (1997) for competitive newsvendors.
• Cachon, G. and S. Netessine. 2004. Game theory in
  Supply Chain Analysis. in Handbook of
  Quantitative Supply Chain Analysis Modeling in
  the eBusiness Era. edited by David Simchi-Levi,
  S. David Wu and Zuo-Jun (Max) Shen. Kluwer.
• Lippman, S. and K. McCardle. 1997. The
  competitive newsboy. Operations Research. 45(1)
  54-65

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Nash Equilibrium

• Nash Equilibrium is characterized by solving a
  system of best responses that translates into the
  system of first-order conditions
• The slope of the best response functions are
  negative, i.e., each players best response is
  monotonically decreasing in the other players
  strategy. (This makes intuitive sense)

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Nash Equilibrium (contd)

• The two best response functions form a best
  response mapping R2?R2 (or in the more general
  case Rn?Rn). One way to think about a Nash
  Equilibrium is a fixed point on the best response
  mapping R2?R2.

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Nash Equilibrium (contd)

• There exists at least one pure strategy Nash
  Equilibrium under the following conditions
• For each player the strategy space is compact and
  convex and the payoff function is continuous and
  quasi-concave with respect to each players own
  strategy.
• However, demonstrating uniqueness is generally
  much harder than demonstrating existence of
  equilibrium.

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The Allocation Game

• Consider a simple supply chain in which a single
  supplier sells to several downstream buyers.
• The buyers are local monopolies (e.g., retailers,
  auto dealers of the same brand) that do not
  compete with each other for customers.
• The supplier has limited (and fixed) capacity and
  sells the product at a fixed wholesale price.
• If buyer orders exceed available capacity (demand
  is high), the supplier needs to allocate his
  restrictive capacity among buyers.
• The allocation policy is publicly known
  (announced).
• How will the choice of allocation policy impact
  buyer behavior and supply chain performance?

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Recommended readings

• Lee, H.L., V. Padmanabhan and S. Whang. 1997.
  Information distortion in a supply chain the
  Bullwhip effect. Management Science. 43(4)
  546-558
• The allocation game is a major cause of the
  Bullwhip Effect.
• Cachon, G. and M. Lariviere. 1999a. Capacity
  choice and allocation strategic behavior and
  supply chain performance. Management Science.
  45(8) 1091-1108
• Cachon, G. and M. Lariviere. 1999b. Capacity
  allocation using past sales when to
  turn-and-earn. Management Science, 45(5)
  685-703. 
• Cachon, G. and M. Lariviere. 1999c. An
  equilibrium analysis of linear and proportional
  allocation of scarce capacity. IIE Transactions.
  31(9) 835-850
• Semiconductor Manufacturing Case
• Mallik, S. and P.T. Harker. 2004. Coordinating
  supply chains with competition capacity
  allocation in semiconductor manufacturing.
  European Journal of Operational Research. 159(2)
  330-347
• Karabuk, S. and S. D. Wu. 2005. Incentive schemes
  for semiconductor capacity allocation a game
  theoretic analysis. Production and Operations
  Management. 14(2) 175188.

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Uniform (fixed) allocation policy

• Under uniform (fixed) allocation, buyer i is
  allocated
• where K is the fixed capacity xi retailer is
  order quantity.
• Assume market price p D q (downward sloping
  linear demand)
• Independent of K, each buyer orders the local
  monopoly quantity
• It is straightforward to show that, under fixed
  allocation, the unique Nash equilibrium has each
  buyer selling minqi, K/2.

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Linear allocation
Further, assume that buyer i orders xil if he
faces low demand situation and xih if he faces
high demand situation. There are some cases in
which identifying pure strategy Nash equilibria
is simple. For example, if Kgt2xih, each buyer
could order xi and be assured of receiving xi. A
buyer could never do better, so this is a unique
Nash Equilibrium. If Kltxih, neither buyer facing
high demand is ever satisfied. To secured the
full capacity, each buyer will try to order K
more that the other buyer. Hence the game reduces
to who can name the largest number and
naturally there is no Nash equilibrium. If
Kltxil, the buyer facing low demand with the
smaller order always receives less than his order
quantity so always has an incentive to raise his
order, thereby destroying any possible
equilibrium. See Cachon and Lariviere (1999c)
for other cases.
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Proportional allocation

• ai(xi,xj) minxi, xi/(xixj)K
• As with linear allocation, there are some cases
  that no equilibrium exists. Symmetric equilibria
  are possible for some cases. See Cachon and
  Lariviere (1999c) for details.
• With any equilibrium under linear or proportional
  allocation, the suppliers expected sales are
  declining in her capacity.
• The supply chain must balance two objectives (1)
  increase the suppliers profit by increasing the
  capacity utilization and (2) increase the
  buyers profit by ensuring that the allotment
  closely matches the buyers true needs. Uniform
  allocation suppresses order inflation, so it
  performs well for the 2nd goal, but may perform
  poorly for the 1st goal. Inflation-inducting
  allocation policies (linear or proportional)
  perform the 1st goal well, and also the 2nd goal
  reasonably well when the order inflation is
  moderate and orderly, i.e., when there exists an
  equilibrium.

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Prioritized allocation

• The Suppliers Decision
• Assume , it is straightforward to show
  that

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Now assume that the buyers do compete for
customers (Cournot competition)

• Assume downward sloping linear demand
• The buyers decisions
• Nash Equilibrium
• The Nash equilibrium has buyer i selling
• and buyer j selling
• (Case 0)

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What if the suppliers capacity K is explicit
knowledge?

• Case 1 (loose supply)
• According to the previous slide, the NE has both
  buyers selling their Cournot quantities.
• Case 2 (tight supply)
• The dominant equilibrium has buyer i selling
• minK , (Dhi-ci)/2 and buyer j selling 0
• under prioritized allocation. (still assume
  )

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The supply chain can be more profitable when
keeping K inexplicit

• Assume
• it can be shown that the supplier is better off
  in Case 0 than in Case 2 ( ) if
  the supply is tight.
• In addition, if implies ,
• it can be shown that the Case 0 overall profit
• is greater than the Case 2
• overall profit
• while

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Multi-period model

• Regular allocation mechanisms.
• Allocation using past sales
• The supplier reserves some capacity for the buyer
  who was the sales leader in the previous period.
• See Cachon and Lariviere (1999b) for details.
• Case A Period 1 Demand Low Period 2 Demand High
• Is it beneficial for the buyers to carry
  inventory over the first period?
• Case B Period 1 Demand Low Period 2 Demand Low
• Is it beneficial for the buyers to allow
  backorders in the first period?

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Another ExampleTwo-Period Allocation, Case A

• Buyers second-period decisions are

Buyers optimal stock level are
given that the first-period allotment is
sufficient to cover s and her first-period sales
quantity.
Further, it can be shown that the marginal
increase in profit from increasing sales in the
second period is greater than the marginal cost
of holding inventory and buyer i would like to
stock a sufficient amount in the first period to
increase second-period sales under the following
condition
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If uniform allocation is used

• In period k, buyer i is allocated
• minxki , K/2(K/2 xkj)
• Suppose q1isi(K/2) gt K/2 (very tight supply),
• the supplier will sell K in each period, and
  buyer i will sell min(2K-(D2-D1)hi)/4 ,K/2 in
  the first period. Sales can be sacrificed in the
  first period so that inventory can be carried
  into the second period, in which the marginal
  value of selling a unity is higher, since in the
  second period neither buyer has an advantage of
  receiving more that K/2 units under an uniform
  allocation mechanism.
• Suppose q2igtK/2gtq1isi(K/2), (moderately tight
  supply)
• buyer i will order q1isi(K/2) in the first
  period and sell K/2si(K/2) in the second period.
  Some of the suppliers capacity might be left
  idle in the first period. The uniform allocation
  may harm the suppliers profitability since it
  will not induce over-ordering from the buyers.

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Conclusion

• Game theoretic models are descriptive models that
  study strategic behaviors of players in a game
  setting.
• In the capacity allocation game, it is important
  for the supplier to choose an appropriate
  allocation mechanism/policy that increases not
  only her capacity utilization but also the supply
  chain profitability and to derive conditions
  under which the policy performs well.