Microeconomics Summary Notes 674vs

Chai Ng 694300

Utility Function

Summarizes what an individual likes/dislikes, underlying preferences. If preferences over bundles of goods satisfy the following assumptions, then there exists a continuous utility function u(x ) that represent these preferences. 1. Completeness: any two bundles can be compared 2. Transitivity: if x> y and y > z , then x> z a. Also depends on an appropriately defined choice set 3. Continuity: if bundle x is better than bundle y, and bundle z is sufficiently close to y, then x is better than z (or if z is closer to y than x , then you’d still prefer x ; can’t be breaks or unexpected changes in preferences) 4. Strict monotonicity: if bundle x contains at least as much as bundle y of every good, and strictly more of at least one good, then consumer would strictly prefer x to y a. Generally, goods providing positive payoffs is not an issue (need to check) b. Allows the assumption of p1 q 1 + p 2 q 2=Y instead of ≤ 5. Convexity: marginally diminishing utility in consumption, therefore consumer would [strictly] prefer bundle z=α x + ( 1−α ) y , for any 0<α <1 a. Strict convexity will provide a unique solution to maximization problem (if not there is an interval of solutions) b. Generally, diminishing utility with consumption not an issue (need to check) Limitations:  Some preferences cannot be represented by a continuous utility function (does not satisfy assumptions)  The utility function is not unique and has varying functions that represent the same preferences (with any positive monotonic transformation, e.g. logs, squares) Example Money Giving Utility Functions:

x=money kept ; y=money given ; 0<α <1

1. Cares about own payoff and the differences in payoff:

u

FS

'

( x , y )=x−α ( x− y )

u FS ( x , y ) =x−α |x− y|

or

2

2. Cares about own payoff and his/her share of the payoff:

uBO ( x , y ) =x−α

(

x 1 − x+ y 2

2

)

3. Cares about moral cost of deviating from social norms:

u¿ ( x )=x−α ( x−x e )

2

Constrained Optimisation Given a constraint (typically a budget set), we are able to solve a utility maximization problem  Assume that utility functions are differential (FOC, SOC) Constrained maximization (find the maximum possible utility that satisfies the constraint) Lagrange method Uses λ to identify the

∂ u with a unit increase in max L ( q1 , q2 , λ ) =u ( q1 , q2 ) + λ (Y − p1 q1 −p 2 q 2)

Y :

q 1 , q2

Steps to solve: 1. Take the FOC and equal to zero 2. Simultaneously solve for q1 and Limitations:

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q 2 (usually a function of Y

and

pi )

1

Chai Ng 694300 1. Not aware that you can consume negative quantities (if impossible, corner solution) Marshallian (uncompensated) demand Turns Lagrange solutions as functions where

q1 ( p1 , p 2 , Y ) and q 2( p1 , p 2 , Y )

Compensation (Expenditure Minimisation problem) How to compensate the consumer to maintain the same level of utility? What is the minimum amount of money the agent needs to guarantee u ´

min p 1 q 1+ p 2 q 2 q1 ,q2

subject ¿ u ( q 1 ,q 2 )=´u

Steps to solve: 1. Solve for

q1 and q 2 through the Lagrange method where L= p1 q1 + p2 q2 + λ( u´ −u ( q 1 ,q 2 ))

2. Calculate the utility and income required (prior price change) 3. Calculate income required from new consumption (given the price change) Hicksian (compensated) demand What bundle a consumer needs to buy to get herself utility way? From Marshallian (uncompensated demand)

u´ in the cheapest possible

qi ( p1 , p2 , Y )=qi ( p1 , p2 , E ( p 1 , p2 , u´ ) )=hi ( p1 , p2 , u´ ) . ´ )= p1 h1 ( p 1 , p2 , u´ ) + p 2 h2 ( p1 , p 2 , u´ ) . Where E ( p1 . p2 , u h ( p1 ,q 1 , u´ ) =( h1 ( p1 , q1 , u´ ) , h2 ( p 1 , q 1 , u´ ) ) Slutsky equation (Income/Substitution Effects)

∂ qi ( p1 , p 2 , E ( p1 , p2 , u´ ) ) ∂ ∂ qi ( p1 , p2 , E ( p 1 , p2 , u´ ) )= hi ( p1 , p2 , u´ )− ∙q 1 ( p1 , p2 , E ( p 1 , p2 , u´ ) ) ∂ p1 ∂ p1 ∂E ∂ hi 1. is the substitution effect: holding utility constant at u ´ , how does ∂ p1 quantity change as the prices change (budget line slope changes) 2.

∂ qi ∙q ∂E 1

is the income effect: holding the price constant (slope of the budget

line), how does quantity respond to the change in income?

Types of Goods Good can change their “properties” depending on particular prices and income 

Normal good:



Luxury good:

∂ qi > 0 (individual consumes more as income increases) ∂Y ∂ qi p 2 q2 ( p1 , p2 , Y ) ∙ >1 (individual spends a larger share of her ∂Y Y

income on the good as income rises) 

Inferior good: o



∂ qi < 0 (individual consumes less as income increases) ∂Y

Often larger impact when the price is lower (since they consumed more units to begin with)

Giffen good: increases)

∂ q1 ( p 1 , p2 ,Y ) ∂ p1

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>0

(individual consumes more as price of good

2

Chai Ng 694300 o

Often has constraints/conditions for the demand to exhibit Giffen properties

Giffen Goods Where to search for a good with Giffen property? When nutrition is a real concern and they are in the “subsistence zone” where they need to meet minimum nutrition levels. There are two types of goods:  Basic good: if households are so impoverished that they consume only the staple good then they are in the “calorie-deprived zone” (steep indifference curves  Fancy good

Game Theory Strategic interaction: my best choice of action depends on what I expect you to do and vice versa How to write a game Strategic form game: 1. Specify players: i=1,2 … N 2. Specify what each player can do:

s i= { s a , sb , … } , resulting in a vector of

strategies s=(s 1 ,.. , s n) that describes what each player has done to determine the outcome of the game s−i is the strategy profile that excludes i ' s pure strategy, therefore a.

s=(s i , s−i ) 3. Specify payoffs/utility functions for each players and their strategies Simultaneous-move (strategic or normal form) games In the example of Penalty Kicks  If the strategy isn’t guaranteed to score, the probabilities of scoring is used as a payoff  Sometimes players don’t want to deviate from the expectation of others Payoff Matrix Player 1 \ Player 2

s 1,b s 1,b

s 2,a s 2,b u1 ( s 1,a , s 2,a ) ,u 2 ( s1, a , s2, a ) u1 ( s 1,a , s 2,b ) ,u 2 ( s1, a , s2,b ) u1 ( s 1,b , s 2,a ) ,u 2 ( s1, b , s2, a ) u1 ( s 1,b , s 2,b ) ,u 2 ( s1, b , s2,b )

Sequential-move (extensive form or dynamic) games Game tree strategies: provides additional information in the definition for who moves when and the histories of the game Game Tree Diagram: How to label information sets, payoffs, strategies and players (in 3 stages)

Strategy in a sequential game is a complete contingent plan of action for each player. At equilibrium, we do not need to distinguish what A thinks B will do, and what B will actually do, because they must be the same. Strategy profile notation, has to write in each stage and branch what the player’s best response is (even if that branch is unlikely to be played out):

( s1, stage1.1 , ( s 1,stage 3.1 , s1, stage3.2 ) ) , ( s 2,stage 2.1 , s2, stage2.2 )

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Chai Ng 694300 Backward induction: beginning from the final branches; prune them until the root (moving the payoffs as you go along) to reach the expected outcome  There are NE’s that are not from backward induction through non-credible threats (deviating from the best response to force player to play a different strategy) o If threatening player can revise action: require players to play NE in every “subgame”, instead of only in the overall game o If threatening player cannot revise action: NE is the appropriate solution concept Subgame: Includes two stages or one action from each player (not a single branch where only one player makes an action!) Information sets: set of nodes where the same player moves; the player who moves has the same set of actions available; a player cannot distinguish histories of the game (will know in what stage, but not which branch)  Subgames cannot break information sets (should behave as it is a new separate game)  Becomes a game with some elements of simultaneous and sequential  Cannot be at terminal nodes (doesn’t make a difference since no one moves after) Strategy profile notation doesn’t need contingency plans for sequential games!

Types of Strategies

Strictly dominant strategy: A strategy that gives player i (strictly) higher payoff than any other of i' s strategies, for any strategy of the opponent Strictly dominated strategy: A strategy that gives player i lower payoff than some other strategy s^i of i , for any strategy of the opponent, is called strictly dominated (eliminate before calculations!) ¿

Pure Strategy Nash Equilibrium (PSNE): A strategy profile s ϵ S is a PSNE if no player can gain by using some other strategy, where the inequality holds (no incentive to deviate / higher payoff): ¿ ui ( s¿i , s¿−i )≥ ui ( si , s−i )

¿

Every player is happy to use their “expected strategy”, where s−i denotes the strategy of other players.  Nash equilibrium need not be efficient! (Expectations can result in a less efficient NE)  Assumption: players do not communicate when they deviate  If additional strategies/options give lower payoff, it will not be chosen (however, this changes the expectations of the other player’s strategies) Systematic way to identify PSNE: 1. Find the best response (or responses) for each player i and each strategy of the opponent, which strategy will provide the highest payoff. B R1 ( s2, x )=s1, x 2. Find where the best responses match each other’s (fixed point): every player i

s i=BR( s−i ) for

¿

Mixed Strategy Nash Equilibrium (MSNE): A strategy profile m ϵ S is a MSNE if for each player i , and every mixed strategy m i of player i , the following inequality holds: ¿ ui (m¿i , m−i ) ≥u i ( mi , m¿−i ) Where m is a vector of probabilities that tell us how often players will play each

strategy where the strategies provide the same expected payoff, players are indifferent between the strategies.

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Chai Ng 694300

β

Probs. Player 1 \ Player 2

α 1−α

s 1,b s 1,b

1−β s 2,a s 2,b u1 ( s 1,a , s 2,a ) ,u 2 ( s1, a , s2, a ) u1 ( s 1,a , s 2,b ) ,u 2 ( s1, a , s2,b ) u1 ( s 1,b , s 2,a ) ,u 2 ( s1, b , s2, a ) u1 ( s 1,b , s 2,b ) ,u 2 ( s1, b , s2,b )

Systematic way to identify MSNE: 1. Calculate the expected utilities for each strategy of player probability β or α u1 ( s 1,a , β ) =β u1 ( s1,a , s 2,a ) + ( 1−β ) u1 ( s1, a , s2,b ) and

i , assume

u1 ( s 1,b , β ) =β u1 ( s1,b , s 2,a ) +(1−β )u1 ( s1,b , s 2,b ) 2. Solve for where expected utilities for each strategy equals, for probability

β ¿∨α ¿ β u1 ( s1,a , s 2,a ) + ( 1−β ) u1 ( s 1,a , s 2,b ) =β u1 ( s 1,b , s 2,a ) + ( 1−β ) u1 ( s 1,b , s 2,b ) u 1 ( s1, a , s2,b ) −u1 ( s 1,b , s 2,b ) β ¿= u1 ( s 1,b , s 2,a ) −u1 ( s1, b , s2, b ) +u1 ( s 1,a , s2, b )−u 1 ( s1, a , s2,a )

Subgame perfect Nash Equilibrium (SPNE): A game that starts at any non-terminal node; A strategy profile is SPNE if it is a NE in the overall game and in each of the subgame.  Any SPNE is a PSNE/NE of the overall game  Not all NE is an SPNE

Congestion Games A game with infinitely many players; when players interact with an “obvious” solution, or provided with an additional option, it changes the agents’ strategy resulting in a less efficient outcome. Common payoffs of players: negative of how many minutes they travel (where y are the proportion of people using a particular route)

x

and

ui ( s i , xs−i )=function of minutes by x How do we find the equilibrium? Where the utilities of the options available equal each other, and no player wants to deviate. 1. Establish general utility functions of the player, for each available route option (as a function of minutes and proportion) 2. Eliminate any strictly dominated routes 3. Set both equations to equal each other and solve for x and total travel time.

Political Competition

1. Players: political parties i={1,2, … n } 2. Strategies: political positioning, as a number in the [0, 1] interval (essentially the same as how parties win voters’ ) a. The parties immediately know who wins (or the probability of winning) 3. How to determine winners? Voters know parties’ positions and has own positions on [0, 1] interval and generally prefers a party that is closest to one’s own position (assume voters vote simultaneously, an SPNE)  Voters: Infinitely many voters, spread evenly across the whole interval [0, 1] (exactly (b−a) fraction of all voters located between points a and b , a
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Chai Ng 694300 

Voter strategy: Therefore

( a+b2 )

vote for

a ,

(1− a+2 b )

vote for

b . Nash equilibrium as no voter can change the outcome and has no incentive to deviate (solution to some non-specified smallest subgame of political positioning):

¿ a if |a−i|<¿ b−i∨¿ b if |a−i|>¿ b−i∨¿ a∨b if |a−i|=¿ b−i∨¿ Vote for ¿ 

Outcome: Assume that parties have probability of ½ of winning each, and parties decide to locate at a=b , each party gets equal number of votes and ties

The winner party is

{

a if a>1−b b if a<1−b a∨b if a=1−b



Styles o Deterministic: party positions determine the winner o Probabilistic: uncertain (only the probability) as party leadership can tilt election in an unpredictable way 4. Payoffs: What the winner cares about? a. About being elected -> losing provides payoff of zero b. About the position of the elected party -> payoff is the same as if it were elected if the winning party is the same position (where x is the position of the winning party) i. Party 0 (a): u0 ( x ) =−x ii. Party 1 (b):

u1 ( x )=−(1−x)

Four Scenarios Winning DW PW

Deterministic voting Probabilistic voting

Position DP PP

DW: (½, ½) is a NE because it is both parties’ best response to each other’s strategies (no incentive to deviate) 

a<

Best response: where share of votes than party)

a

1 , B R b ( a ) ϵ (a ,1−a) as b will have a larger 2

(essentially to be in the interval of the opposing

PW: (½, ½) is an NE because its where 

B R a ( b )=a

and

B R b ( a )=b

1 , B R b ( a )=a+ 0.01 to maximize its 2 1 probabilities of winning (or if a> , B R b ( a )=a−0.01 ) 2 Best response: where

a<

DP: (½, ½) is an NE, as no party as incentives to deviate because it loses the election. Best responses  

1 1 a< , B R 0 ( a )=1−a−0.01 and if a> , B R 0 ( a )=lose 2 2 1 1 For party 0, if b> , B R 1 ( b )=1−b+ 0.01 and if b< , B R 1 ( b )=lose 2 2 For party 1, if

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Chai Ng 694300 PP: (0, 1) is the NE as it is the strictly dominant strategies where parties care about their positions and voting is probabilistic. Identifying best responses require setting up a maximization problem (as it becomes expected utility from winning/losing). Best responses:

max E u0 ( a , b )=(−a ) a

¿

( a+b2 )+ (−b ) (1− a+b2 )

b 2−a 2 −b 2 

Party 0 can’t control b, therefore, for any b,



Similarly for Party 1, for any a,

B R 0 ( b )=0

B R1 ( a ) =1

Median voter theorem: parties locate at the position of the median voter (assumes that voters are distributed uniformly on [0, 1]). However, this invalidates the conclusion:  Not everyone has to vote (more important to get people to vote)  Elections are structured as multiple contests (median voter may not apply to the whole country)

Standard Auctions Two key areas of auction theory: 1. Bidding strategies: How should bidders bid to maximize their expected payoffs? a. bi ( v) is the bidding function that tells you what you should bid for every possible valuation

vi

(may depend on your best guess about the

distribution of others F j (v ) , as well as the number of other bidders) b. Assume that other players are playing a random (mixed) strategy c. Payoffs are assumed to be v i −bi , and zero if bidder left the auction 2. Auction design: How should sellers design the auction to maximize their expected revenues? (or efficiency in a government example) a. Simplicity sake, we assume that the distribution is uniform on [0, 1], for second-highest order statistics (

for sample ¿ :

n−1 ), the expected n+1

revenue of the auctioneer is: 2 3 4 5 6 1/3 ½ 3/5 2/3 5/7 b. Preference for second-price auction for single unit private value auctions is because the strategy is easier to understand and explain that bi ( v )=v c. Governments care about efficiency above all else (because if item is allocated to a company that does not have the best use for the asset, the money or potential gain is lost forever Environments: Standard: Private values, you have your own valuation that is unaffected by others (e.g. computers)  Each bidder i' s valuation is a random, independent draw from some distribution with cumulative density function Fi (v) that represents’ the bidders’ and auctioneer’s best guess about the valuations. v is the actual amount willing to pay for the object after seeing it. Different distributions for different bidders.  Results in Revenue Equivalence Theorem: if bidders are risk neutral and their valuations are independent, the seller’s expected revenues are the same in these four standard private-value auctions o Distribution of revenues and bidding strategies are different, which the seller may care about

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Chai Ng 694300 



More likely to receive low revenue form the second-price auction than from the first price auction, but chances to have higher revenues (higher variance) Seller will never get more than 2/3 in the first price auction

Common Values: The value of the item is the same for you as others (details matter, e.g. selling a painting/antiques)  Winner’s curse: incorrect conditioning of his expected value will often result in the winner overpaying because: o The winning bid exceeds the value of the auctioned asset such that the winner is worse off in absolute o The value of the asset is less than the bidder anticipated, so that the bidder as a net gain but will be worse off than expected (winning has a negative signal that your valuation is too high)  How to confirm this? E(v i whenbi >r ) o Expected (true) valuation (assume best response):

o 



∑ ( probability of valuation )∗(average of valuations) for all scenarios Expected payoff: ∑ ( probability of occuring )∗( payoff ) of all scenarios

Difficult to calculate the optimal bid, but bidding changes when: o There are more bidders, you must bid less aggressively (more likely to lose the object, but less likely to get a lemon) o There is a famous expert, you must bid less aggressively (same as above) Ways to overcome this and considerations: o How valuable the information is for others? o How easy it is to obtain information?

Open vs. Sealed bids: Ascending price auction (open): more difficult to signal own bidding strategy to the other bidder and pay back to combat inefficiencies in common value auction Early vs. Late bids: Only important in a “sealed-bid” style environment where there is an advantage in concealing superior information.  Bidding early results in the standard second-price auction.  Bidding late means rival has no time to respond, but bid will be accepted with only probability α

b2 wins b2 loses 2 b1 wins α ( 1−α ) α b1 loses ( 1−α ) α ( 1−α )2  How to compare? Identify E ( u1 (late ) )− E ( u1 ( early ) ) , where E ( u )= probability of winning∗(v i −price) 

Higher valuation bidders would prefer to bid early and get the object with high probability

Formats: English (ascending): bidders submit their bids sequentially as open outcries, with the highest bidder winning  Where r is the current highest bid, a bidder i can bid bi=r +0.01 or leave the auction, the best response (and weakly dominant strategy) is:

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Chai Ng 694300

{

bid b i=r +0.01 if v i >r + 0.01 leave the auctionif v i
Ascending clock auction: Price is announced, and bidders can either accept (price p+increment is announced) or leave the auction; if there is no more bidders at the new increment, object is allocated randomly among bidders (accepted price); if there is one bidder, she gets the object; if more than one bidder, the process repeats  Provides additional information (signals) while the auction is run; able to revise your bid  Auctioneers’ revenues are higher in ascending price because the information of others is revealed and they are able to bid more aggressively  More efficient than second-price auction, combat inefficiencies in common value auction  Similar to Amazon (additional 10 mins for each extra bids so that others would have a chance to react), resulting in no major difference in the bidding strategies of “common value” and “private value” goods as bidders have no incentive to hide their information Second-price sealed-bid: Bidders submit their bids simultaneously, with the highest bidder winning and paying the second highest bid  Weakly dominant strategy to submit a bid bi=v i as payoff is v i −r , where r is unknown o If i loses the auction, any other strategy would either make i lose the auction, or lose money if i wins the auction o If i wins the auction, any other strategy would either give i the same payoff, or make i lose the auction (because i pays the secondhighest price) o Bidding bi=v i gives i at least the same payoff as any other strategy and sometimes a higher payoff (consider the scenarios of where

v i >¿< ¿=r



Therefore, the bidder with the highest valuation wins and auctioneer receives the

n−1 n+1

expected revenue of 

eBay as bidders can jump the bid up at the very last second, so that others cannot react (also bidders prefer to “gamble” and get the object at a lower price at a lower probability, than to get the object at a very high price with a high probability)

Dutch (descending): Auctioneer starts with a high price and reduces the price, with the first bidder who accepts a price winning  

( 1n ) v

bi ( v )= 1−

i

similar to first-price sealed-bid is the ideal bidding strategy

Once price is below your valuation, waiting increases your payoff (if you win) but lowers the probability that you win

First-price sealed-bid: Bidders submit their bids simultaneously, with the highest bidder winning and paying the second highest bid 

NE is where o

bi ( v )=

v 2

for 2 players

Require a maximization problem setup: what is the probability of winning with bid

x ? Or what is the probability that

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x

is higher than

b2=

v2 2 9

Chai Ng 694300 (uniform distribution of [0, 1]), therefore the probability is

2 x , where x ≤

1 2

(to ensure probability is not larger than 1)

max [2 x ∙ ( v 1−x ) ] x

Take the FOC and set to zero results in

o

Bidders best response strategy, as the number of players increase, following the pattern of



v1 2

o

( 1n ) v

bi ( v )= 1−

i

(the more bidders there are,

the more aggressively they bid Auctioneer gets the expectation of the highest-order statistic (

n )and plug that expected valuation into the bidding function n+1 n 1 n−1 1− = n+1 n n+ 1

for sample ¿ :

( )

Multi-unit Auctions: Spectrum Auction, “Treasury” Auctions, coal or offshore petroleum exploration rights.  Assume two identical units, and three bidders (know their own valuation but not the valuation of others)  Assume using the second-price sealed-bid auction  If bidders demand more than one unit, weakly dominant strategy for the first unit, but incentive to bid lower on additional units  Complements/substitutes: difficult to efficiently allocate multi-unit goods. Package bidding allow bidding on both individual items, as well as a collection of lots (more efficient, but may discourage small bidders to participate). There are n bidders, each interested in buying 2 units of good at price $1, with n units for sale (but demand for 2 n ). n highest bids determine the allocation and n+1 -highest bid determines the price.  Zero-revenue NE: Each bidder bids $1 for unit 1 and $0 for unit 2. Payoffs: $1/bidder, $0 for seller. No incentive to deviate as if bidder bids $1 for the second unit, the n+1 -th bid is now $1 (new price) and the payoff is zero regardless if the bidder gets the object or not.  “Good” NE: Each bidder bids $1 for both units. Payoffs: $0/bidder, $1 for seller. No incentive to deviate as they can’t change the price ( n+1 -st bid of $1) Sequential multi-unit auction: Auction unit 1 first, then unit 2 next. In the last auction, everyone should bid true valuation. However:  Highest bidder would not necessarily leave after the first round (prefer to let second-highest bidder win first round and leave, and compete with lower bidders, but everyone is making these calculations so hard to compete optimal bidding strategy in second round)  Difficult to expand Simultaneous multi-unit auction:  Separate auctions for each unit: Difficult because bidders can either win both units (doesn’t want both), or have an unallocated unit, or receive low revenues, or have bidders revaluating their strategy to bid lower (NZ Auction)  Each bidder pays the next bidder’s bid: if b1 >b 2> b3 , then bidder 1 pays b2 and bidder 2 pays b3 (Bidder 1 will consider bidding lower than b2 and pay

b3

instead), as the advantage of second-price auction is lost:

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Chai Ng 694300 Higher bid (higher probability of winning) vs. Lower bid (lower expected price) Each winner pays the same price: bidder 1 and 2 pays b3 . Bidders changing their bid will change whether they win and when they win, but does not change the price paid (similar to single-unit second-price auction, uncertain effects on revenue) o n highest bidders get n object and pay n+1 -st highest price Simultaneous ascending auction (open): bidding occurs over many rounds o only stops when no license receives a new bid (allows bidders to switch to other licenses if the license they are bidding for becomes too expensive) o if bidders bid too little, they are restricted from bidding in the next round (forces bidders to bid, instead of “waiting out” and bidding late) o





Non-standard auctions Where there are no fixed seller or buyer, and no money changing hands. American football: How do we get both teams to be indifferent in how the coin lands (determining who wins the possession) ¿

¿

We want to find where p ( x )=q ( x ) where probability of team A winning is the same whether A or B gets the ball, and likewise for team B who will not care  p(x) the probability of A winning if A has possession (probability of B winning, if A has possession is 1− p(x ) )  q( x) the probability of A winning if B has possession (probability of B winning, if B has possession is 1−q ( x ) ¿  x yardage for team A’s end-zone ¿

Descending distance auction: Assume NE when x=x where neither team wants the ball and there is coin flip (indifferent); however unrealistic to assume that teams know each other’s probability of winning ( p(x) and q( x) )  Becomes similar to a common value auction, where teams that haven’t dropped out signals that own valuation is too low/high Average-distance sealed-bid auction: Winner is lower bidder, with starting distance being the average of the two; how do we know if the teams will bid close to their own true valuation?  Team A’s utility from winning the possession pushes the bid up:

p ( x ) u A ( Win the game ) + ( 1− p ( x ) ) u A ( Lose the game) 

Team B’s utility from losing and the other team’s possession pushes the bid down:

( 1− p ( x ) ) u B ( Winthe game )+ p ( x ) uB ( Lose the game) Bilateral Trade Assume that seller also has a valuation for the item and is unwilling to sell if

bi < v s (no

v s< v B ) p ( v S , v B )= price (if transaction

auction/possible mechanism to induce efficient trade where 

Pricing function (utility/revenue/payoff function): happens)

Myerson and Satterthwaite Impossibility Result: 1. Seller can’t sell an item that they don’t have (physical constraints) 2. Information constraints can also prevent the sale of an item

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Chai Ng 694300 a. Seller/buyer can pretend they have a valuation different from their true valuation (seller wants higher; buyer wants lower) uS ∨B ( What S∨B say|Who S∨B really is¿ b. c. Cannot guarantee all efficient trades will happen if no truthful reporting Individual rationality constraints/conditions (because trade is voluntary): p ( v s , v B ) ≥ v s , otherwise seller won’t sell   

p ( v s , v B ) ≤ v B , otherwise buyer won’t buy p ( v s , v B )=0 , trade shouldn’t happen in this case

Incentive compatibility constraint: What if the sellers/buyers pretended they had different valuation? (1) Find expected utility if the scenarios if they pretended/if they hadn’t (2) set true valuation to ≥ pretend valuation utility (3) calculate pricing functions for the ones you know  For the seller p ( 0,1 ) ≥ 0.8 , but for the buyer p ( 0,1 ) ≤0.2  We need incentive compatibility for players to report truthfully Mechanism 1: Seller makes an offer If seller’s valuation is either v s=0 or

v s=0.9 , likewise buyer valuation is either v B=0.1 or v B=1 . NE: Both v S=0 and v S=0.9 will offer p=1 (only sell where v B=1 ) which provides the highest expected revenue Mechanism 2: Seller makes two offers In equilibrium p1> p 2 , every buyer will reject the first offer and wait for in Mechanism 1 principles Mechanism 3: Buyer makes an offer Similar to Mechanism 1, Buyer is better off buying at payoffs

p=0

p2 , resulting

with higher expected

Mechanism 4: Second-price auction Simultaneous naming of valuation, if v B >v S

then trade happens where pB =v S and pS =v B (but where would the difference in money come from? Sometimes government

is interested in efficient trade)  Can get money from agents themselves, before they know their own valuations (g a contract before that to put money into escrow )  Better to contract before parties have their information

Adverse Selection Assume that only one side has information, but this information is relevant for the other party (buyer does not know his valuation for the object but the seller will) Hostile takeover: Company knows better than acquirer their valuation. What price should the company set?  Investors’ value function v I =ap , therefore willing to buy only when

v I =aE ( p ) =p

 

E( p) is the expected price/value of the company lower bound + p which is often ) 2 Company will not offer p , if it is worth more than p and the investor knows (where

this; understanding that the expected value on a random variable distribution is now p/2 from [0, p ] If investor knows more information, narrows the band of valuation and investor is more willing to pay a higher price

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Chai Ng 694300 Insurance: Customer knows better about her health/driving habits that will determine the cost to the insurance company  Each agent comes with v i which is the expected value of insurance claims (cost to insurers) distributed uniformly on [0, 1] and value the policy at 1.2 v i (risk adverse)  Only agents whose 1.2 v i ≥ p will buy insurance  Total cost to insurer is the cost/person * # of people insured ( E ( v i )∗Fraction of insured people¿ ; Total revenue is p∗Fraction of insured people ; Total profit is the revenue – cost (insurer will price when it’s profitable) o Only a small fraction of people are insured, as only high-cost agents are willing to sign for insurance driving up the price  Solution? Forcing “desirable” customers to sign up Unobservable quality: such as internal damage of a car that affects the quality of the car Solutions: 1. Forcing desirable customers into the market 2. Signaling: High-quality market will be efficient a. E.g. Warranty for a car, “cheap” for a high-quality item but “expensive” for a low-quality item; resulting in only high-quality cars coming in with warranty

Random Thoughts   



     

Every model has its setup, assumptions (that are able to be tested) and is able to make predictions; should not always conform to reality Know his model types and know the assumptions/understanding which will be tested Math Notation: Bold letters will always stand for a vector (e.g. x=( x 1 , x 2 ) and

y=( y 1 , y 2 ) will represent in the bundle x=(1,3) , there is 1 unit of good 1 and 3 units of good 2) Experiments/Situations: invented environments/situations to test the validity of models o Utility prediction: (1) set assumptions/limits, (2) FOC=0, (3) interpret relating back to (1) assumptions and limitations Selfish behavior: okay to assume because most of the time it provides accurate/simple predictions Proof by contradiction: proofing technique that assumes the statement is wrong, then prove that the condition of the theorem is not satisfied Simplified problems are used when the initial problem is too hard (assuming it satisfies the same constraints) Consumer theory informs about details of field experiments, who to target, how to design, etc. In exam, don’t try and find MSNE for 3x3 (or more) game, you can be sure to rule out strictly dominated strategies before beginning calculations If something doesn’t work in a simple environment, it won’t work in a more complicated environment x−startpoint 2



Probability of winning on a uniform distribution is

  

In second-price auction, if both ties, they pay their bids (which is equal anyways) Expected price of the rival on a uniform distribution is the midpoint Deg auctions in non-money environments o What is the object being allocated and how? (allocation rule)

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Chai Ng 694300 o

Is there a variable that can serve the role of money (payment rule) that is (noted as p(v s , v B ) )  (1) sufficiently divisible?  (2) sufficient range to make participants wish not to win the auction?

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