MBA 201A.1a,2a Section 6: Auctions

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MBA 201A.1a,2aSection 6Auctions Firm
Competition

• Dylan Minor

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Overview

• Auctions
• Cournot vs. Bertrand vs. Stackelberg-
  illuminations and a really long problem
• Repeated Games (time permitting)
• QA

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Class Concepts Auctions

• Auctions
• One-time auction for a single item
• English Auction Ascending bid auction. The
  item goes to the last party to bid.
• Strategy When to drop out ? What is your WTP?
• Dutch Auction Descending bid auction. The item
  goes to the first party to bid.
• Strategy When to start bidding ? Survey the
  competition (want to bid the value of bidder just
  below you)
• First-Price Sealed Bid All interested parties
  submit a price. The item goes to the party with
  the highest bid, at the price he/she submitted.
• Strategy What price to submit ? Survey the
  competition (want to bid the value of bidder just
  below you)
• Second-Price Sealed Bid All interested parties
  submit a price. The item goes to the party with
  the highest bid, at the price of the second
  highest bid submitted.
• Strategy What price to submit ?What is your WTP

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Class Concepts Auctions

• Auctions
• So, in summary
• Ideally pay bid your WTP for 2nd price, WTP to
  pay of next bidder .01 in 1st price

Example 1 two bidders, you with value 1 and the
other with value .5, what do you bid under 1st
price? Second Price? What do you pay?
Example 2 two bidders, you with value .5 and
the other with value 1, what do you bid under
1st price? Second Price? What do you pay?
Example 3 two bidders, you with value .5 and
the other with unknown value uniformly
distributed from 0 to 1, what do you bid under
1st price? Second Price? What do you pay (in
expectation)?
Answers Example 1(.5,1,.5) Example
2(0,.5,0) Example 3(.25,.5,.25 expected if
win .125 unconditional expectation (50 chance
of win X .5))
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Bertrand, Cournot, and Stackelberg

• You always want to ask is the (direct)
  competition variable PRICE or QUANTITY and is the
  decision SIMULTANEOUS or SEQUENTIAL

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Bertrand, Cournot, and Stackelberg

• A Very Long Example
• P10-Q, MC1, 2 Firms
• What is the Bertrand Equilibrium? What if MC11
  and MC22?
• What is the monopoly price/ quantity/ profit?
• What is the one shot payoff (and quantities
  provided) of deviating from collusion of
  providing monopoly quantity (round down to 2
  being the split monopoly quantity)? What if
  instead the firm deviates by lowering price (in a
  one shot)?

Both price at MC1, supply a total of 9 and make
0 profit. Firm 1 prices at 1.999, supplies
roughly 8, roughly 8 profit (vs. only 4 profit
if sold at 2 and shared the market with firm 2).
Note the threat of entry maintains the price.
Total Q4.5, P5.5, profit20.25(5.5-1)x4.5
Residual demand for firm 1 (deviator)
P10-q1-28-q1 MR8-2q11q13.5P10-2-3.54.
5 Profit firm 1 (4.5-1)x3.512.25 and Profit
firm 2 (4.5-1)27 vs. each having 10.125
each with collusion However, if firm one lowers
the price by a tiny amount, it can take all the
business, making roughly 20.25 vs. 12.25 by
deviating via quantity. Meanwhile, firm 2 had
produced 2.25 units for 2.25 thinking the firms
were colluding, meaning it lost 2.25. Which
deviation would you do- quantity or price?
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Bertrand, Cournot, and Stackelberg

• A Very Long Example
• P10-Q, MC1, 2 Firms
• What does the game look like via a matrix (assume
  10 collude, 20 deviate (or -2))?
• Which strategies survive iterated elimination of
  strictly dominated strategies? Which strategies,
  if any are dominated, and which, if any are
  dominating? What is the Nash Equilibrium?
• What discount rate d is needed to maintain
  collusion (under a Grim trigger)?

Answer Collude is strictly dominated, Deviate is
dominating, and (Deviate, Deviate) is our sole
Nash Equilibrium- refer to the last slides on
effortlessly finding equilibria via the
underlining method
Answer we need expected value of collude
expected value of deviate 10/(1-d)20(d/(1-d)x0
10/(1-d)20 d1/2, then we have tacit
collusion, which is equivalent to an r(i.e., d1/(1r))
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Bertrand, Cournot, and Stackelberg

• A Very Long Example
• P10-Q, MC1, 2 Firms
• What is the Cournot equilibrium?
• What is the Cournot equilibrium if MC11, MC24?
• Questions?
• REMEMBER an equilibrium is denoted by the pair of
  outcomes (i.e., here the two optimal quantities
  produced) not payoffs!

Answer P10-q1-q2 profit for firm
1(10-q1-q2)q1-1q1 Take the derivate and set 0
q1(9-q2)/2. Note this is equivalent to using
MRMC, which here is 10-2q1-q21. Since, firm 2
has the same MC, we have q2(9-q1)/2. Solving for
q1 by plugging in q2 gives q14.5-2.25.25q1
q13q2 P4, profit for each firm(4-1)39.
Answer Same exact method as above yields q14,
q21 P5 profit firm 1 (5-1)416 Profit
firm 2 (5-4)11- note total profit went to
17. In the limit (of firm 2s increasing
marginal cost), firm 2 is pushed out of business,
giving firm1 monopoly position.
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Bertrand, Cournot, and Stackelberg

• A Very Long Example
• P10-Q, MC1, 2 Firms
• With the Stackelberg game, one firm enters first
  providing quantity and then the second firm
  provides a best response quantity.
• How do we solve this problem- i.e., find the
  equilibrium?

Answer We begin as before in needing to
calculate the best response functions. We
immediately know we have q2(9-q1)/2. However,
now since firm 1 goes first, it is solving
profit (10-q1-((9-q1)/2)))xq1-1xq1. We get
10-q1-4.5-10 q14.5. Take the derivative and
set it equal to zero, then solving for q1,
yielding q14.5 q22.25 price3.25. Firm 1
profit 10.125 versus firm 2 profit 5.0625.
What is the first mover advantage?
10.125-5.06255.0625.
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Bertrand, Cournot, and Stackelberg

• A Very Long Example
• P10-Q, MC1, 2 Firms
• How do we model the Stackelberg game as a game
  tree (assuming each player ONLY does their best
  response)?

Answer see above. Round off monopoly profit to
20 and Stackelberg to (10,5) for firm 1 and
firm 2. Our unique Nash Equilibrium is firm 1
enters and 2 plays the Stackelberg. Note Cournot
is not an option for firm 2 because it is
entering 2nd.
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Questions?
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See you Monday 6-8 pm in Andersen for final
review
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Class Concepts Repeated Games

• Repeated Games
• Repeated Games are those with simultaneous
  decisions over an infinite number of interactions
  or no known ending point.
• Games with simultaneous decisions and a finite
  number of interactions are sequential games.
  They should be solved using backward induction,
  and not the strategies discussed here.
• Strategies for Repeated Games
• Goal Sustain price moderation (and higher
  payoffs) to both firms, even though this is not
  an equilibrium solution to the simultaneous game
• Grim Strategy Price at the optimal industry
  price. If an opponent cheats and prices lower,
  punish them by pricing at marginal cost forever.
  
• t-period Trigger Strategy Price at the optimal
  industry price. If an opponent cheats and prices
  lower, price at marginal cost for the next t
  periods. Then return to the optimal industry
  price.
• Tit for Tat Strategy Price at the optimal
  industry price. If an opponent cheats and prices
  lower, then price at marginal cost in the
  subsequent period.

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Class Concepts Repeated Games

• Repeated Game Strategy Evaluation
• Let High Profits H, Low Profits L, Cheat
  Profits C
• Discount r
• Grim Strategy ? If both choose this strategy, is
  it an equilibrium? Will parties want to cheat?
• Profits from Both Pricing High H/(1r)
  H/(1r)2 H/(1r)3 H/r
• Profits for a Cheater C/(1r) L/(1r)2
  L/(1r)3 (C L/r)/(1r)
• If H/r (C L/r)/(1r), then both parties
  pricing high using the Grim Strategy is an
  equilibrium. Otherwise, parties have incentive
  to cheat, and pricing high will not be
  sustainable.
• t-period Trigger Strategy ? If the Grim strategy
  wont work, this wont work. If the Grim
  strategy does work, how many periods will you
  need to punish your opponent to get them to price
  high/not cheat?
• Profits from Both Pricing High H/(1r)
  H/(1r)2 H/(1r)t1
• Profits for a Cheater C/(1r) L/(1r)2
  L/(1r)3 L/(1r)t1
• If the profits from both pricing high for the
  first t1 periods are better than the cheating
  profits for the same periods, then the t-period
  trigger strategy is an equilibrium. Otherwise,
  parties have an incentive to cheat, and this
  strategy will not be sustainable.

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Class Concepts Repeated Games

• Knowledge Check
• Two firms can receive 8M in profits each if they
  both price high. If one of them decides to price
  lower, that firm will receive 15M in profits
  that period and the other firm will receive 0 in
  profits. If both firms price low they will
  receive 4M in profits each. Use a 10 discount
  rate.
• 1) Is the grim strategy effective?

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Class Concepts Repeated Games

• Knowledge Check
• Two firms can receive 8M in profits each if they
  both price high. If one of them decides to price
  lower, that firm will receive 15M in profits
  that period and the other firm will receive 0 in
  profits. If both firms price low they will
  receive 4M in profits each. Use a 10 discount
  rate.
• 1) Is the grim strategy effective?
• Profits from Pricing High 8M/0.1 80M
• Profits from Cheating 15M/1.1 (4M/1.1
  4M/1.12 )/1.1
• Profits from Cheating (15M 4M/.1)/1.1
  50M
• Yes, the grim strategy for both parties is an
  equilibrium.
• 2) Is the t-period trigger strategy effective for
  t 2? For t 3?

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Class Concepts Repeated Games

• Knowledge Check
• Two firms can receive 8M in profits each if they
  both price high. If one of them decides to price
  lower, that firm will receive 15M in profits
  that period and the other firm will receive 0 in
  profits. If both firms price low they will
  receive 4M in profits each. Use a 10 discount
  rate.
• 2) Is the t-period trigger strategy effective for
  t 2? For t 3?
• For t 2 ? No, the t-period trigger strategy is
  not an equilibrium for the game
• Profits from Pricing High 8/1.1 8/1.12
  8/1.13 19.89M
• Profits from Cheating 15/1.1 4/1.12
  4/1.13 19.94M
• For t 3 ? Yes, the t-period trigger strategy is
  an equilibrium
• Profits from Pricing High 8/1.1 8/1.12
  8/1.13 8/1.14 25.36M
• Profits from Cheating 15/1.1 4/1.12
  4/1.13 4/1.14 22.68M
• So, you should price low for at least 3 periods
  to punish your opponent for cheating to make
  price moderation behavior the dominant strategy.