• Groves-Clark Mechanism: gets individuals to truthfully to reveal their preferences for public good
• 1- Individuals report their value for the bridge vi
• 2- add up reported values.
• 3- only build if Sum of Reports – Cost of Bridge >0
• 4- if the individual’s value was decisive (made project go ahead)
- i.e. Sum of Others’ Reports < Cost of Bridge < Sum of all Reports
• Then we charge the individual: Charge= Cost of Bridge – Sum of others’ reports
• U is the sum of the other’s reports. v is my value
• If U>cost then project is already going ahead. Doesn’t matter what I do, so bid truthfully and not
charged
• If U+v > Cost > U: now we are decisive. We value at v but report a u such that U+u> cost .
• This will result in us getting a positive utility of v-(cost-U)
- as we are charged for being decisive, charged this amount (C-U)
• This utility is independent of what we reported.
• But, if we report a u that is too low i.e. U+u<cost then the project doesn’t go ahead and we get
utility 0
• It is weakly dominant to tell the truth as this ensures I get positive utility even though I pay for
decisive
• Summary: optimal to tell truth, only pay when decisive (but person always pays less than value),
payments< benefits so need funds to build project. As population grows less of a problem with
excess revenue.
Auctions
• When monopolist doesn’t know its demand function, it may use an auction. Competition among
buyers gets them to tell you the true demand. The monopolist solves a mechanism design
problem
• First-price Sealed Bid: secret bids, highest bidder wins and pays his bid (static)
• Second-Price Sealed bid: highest bidder wins but pays second highest bid
• Dutch Auction: start with a high price and slowly reduce it with a clock. First buyer to press his
button stops the clock and gets object at the price on the clock. Dynamic and similar to 1st price
• English Auction: start with low price, auctioned slowly increases it.
- Last buyer left gets the object at the price where the penultimate buyer was eliminated. Typical
tv one
• Double Auction: K seller with 1 unit each to sell and many buyers. Sellers and buyers
simultaneously submit bids. The K highest bids get an object.
- The price is determined by an average of the K and K+1 highest bids.
- Sellers with bids above this price don’t sell their object, sellers with bids below this price
do sell.
• Single Unit Auction Model: one seller, N buyers who have a value vi∈[0,1] for the object
• If bidder wins at price p, utility= vi-p else 0 if lose
• The realisation of (v1, v2,..., vN) determines the seller’s demand curve.
• Independent Symmetric Private Values: vi is drawn independently from density f(v) on [0,1]
• Independent Private Values: vi is drawn independently from the density fi(v) on [0,1]
- i.e. player i and j have different distributions of values
• Correlated Private Values: The vector (v1, v2,..., vN) is drawn from a full support density on [0,1]N
- vi and vj are related
• Common Values: v1=v2=...= vN but players do not know what v is. They have
private information on it. Never have different values hence probability is 0
• We consider Independent Symmetric Private Values.
• Players only observe their own value and then bid.
• Player’s strategies are bidding functions that maps the player’s value to a bid bi(vi)
• We only consider equilibria where bids are strictly increasing functions of values
• See older notes for 2nd price auctions, similar ideas apply for English Auctions
, Designing Economic Systems
• A seller can build the project and the buyer can use it. If the buyer and seller agree on terms, the project
goes ahead i.e. price buyer pays to seller
• The project is ‘easy’ for the seller with probability 1/2 and low cost C=35 and the project is ‘hard’ and
expensive at C=60
• Buyers have low value V=40 and high value V=60, both with probability 1/2.
• Sellers don’t know buyers’ values and vice versa. Can’t force participation
• We plot the gains from trade
• If there was complete information, trade is efficient 3/4 of the time
• Each call has a probability of (1/2)(1/2)= 1/4 so expected gains from trade are 1/4(5) +
1/4(0) +1/4(5) + 1/4(30) =10
• Myserson Satterhwaite Theorem: there is no bargaining procedure under which the project
goes ahead even when it’s efficient to do so.
• We organise a system of trade where we get the buyer and seller to tell us their values
and then choose a price that shares the gains from trade equally
• If the seller announces he is a low cost then the project always goes ahead
- 1/2 the time at a profit of 2.50 (37.5-35), and half the time at profit of 15 (50-35), for an
average of 8.75
• If the seller is low cost but pretends to be high cost then the project gets built 1/2 of the
time at a profit for the seller of 62.50-35= 27.5
• So expected profit from lying is 1/2(27.50)=13.75
• The seller makes a higher expected profit from lying so the low-cost seller will lie. This scheme fails
• As long as values are private information, there is no mechanism that gets players to tell the truth and
builds the project whenever it should be built (no efficient way to organise trade)
• it is impossible to get the buyer and the seller to truthfully reveal their type here
• To get a high-cost seller to trade with a high-value buyer, we must promise them a price of at least 60 half
the time. At an efficient outcome he will end up building the project 1/2 the time
• If the seller actually had low costs, he could lie and pretend to be high cost to get at least 1⁄2(60-35)=12.5
• The real mechanism must give the seller expected profits of at least 12.5 when he is low-cost to ensure
he tells the truth.
• The seller expects to earn at least 12.5 at least 1⁄2 the time (when low cost) and 1⁄2 the time gets 0
• The seller’s expected profit is therefore 6.25. We must give at least this surplus from trade to the seller
• The buyer can pretend to have low value so he must have expected profit of at least 6.25, independent of
the mechanism. We must give both people incentives to tell the truth
• But the expected gains from trade were only 10 but we need 12.5 (6.25+6.25)
• It’s impossible to arrive at an efficient outcome using bargaining/negotiation when there’s private info
Mechanisms and the Revelation Principle
• To describe a problem, we say: there are individuals in the economy. Set of possible allocations of social states: x ∈X.
• Each individual has a utility function ui(x,θi), depends on social state chosen and on their private type/ knowledge θi
• Individual i’s type is selected from a set Θi by a known probability distribution e.g. before it was 1/2 low cost
• A mechanism describes how planner organises- receive info from agents as signals, aggregate then make decision
• A mechanism provides each individual in society a set of signals Si that they can send . It also announces a policy
rule γ which turns the signals into an allocation x
- e.g. message is how much you value bridge. Policy rule is whether to build and who pays for it
• Process: set up mechanism, individuals observe their utility and type. The messages sent depend on their type so we
denote this as si*(θi) . This is a game because the messages one individual sends depends on what others do.
• Once all the messages are sent the allocation gets decided.
• The equilibrium is denoted as γ(s1*(θ1),..., sn*(θn)) =x
• The policy rule maps types to a decision
• We can write this as Γ(θ1,..., θn) = x
- We say it is possible for the planner to implement the decision rule Γ(θ1,..., θn) = x
• e.g. decision rule is to oblige the buyer and seller to trade iff buyer is high value and seller is low, this is possible
• Revelation principle: if strategies si*(θi) are an equilibrium in dominant strategies, we only need agents to tell us their
type Si =Θi . Don’t need any other messages
• 1- Individuals report their value for the bridge vi
• 2- add up reported values.
• 3- only build if Sum of Reports – Cost of Bridge >0
• 4- if the individual’s value was decisive (made project go ahead)
- i.e. Sum of Others’ Reports < Cost of Bridge < Sum of all Reports
• Then we charge the individual: Charge= Cost of Bridge – Sum of others’ reports
• U is the sum of the other’s reports. v is my value
• If U>cost then project is already going ahead. Doesn’t matter what I do, so bid truthfully and not
charged
• If U+v > Cost > U: now we are decisive. We value at v but report a u such that U+u> cost .
• This will result in us getting a positive utility of v-(cost-U)
- as we are charged for being decisive, charged this amount (C-U)
• This utility is independent of what we reported.
• But, if we report a u that is too low i.e. U+u<cost then the project doesn’t go ahead and we get
utility 0
• It is weakly dominant to tell the truth as this ensures I get positive utility even though I pay for
decisive
• Summary: optimal to tell truth, only pay when decisive (but person always pays less than value),
payments< benefits so need funds to build project. As population grows less of a problem with
excess revenue.
Auctions
• When monopolist doesn’t know its demand function, it may use an auction. Competition among
buyers gets them to tell you the true demand. The monopolist solves a mechanism design
problem
• First-price Sealed Bid: secret bids, highest bidder wins and pays his bid (static)
• Second-Price Sealed bid: highest bidder wins but pays second highest bid
• Dutch Auction: start with a high price and slowly reduce it with a clock. First buyer to press his
button stops the clock and gets object at the price on the clock. Dynamic and similar to 1st price
• English Auction: start with low price, auctioned slowly increases it.
- Last buyer left gets the object at the price where the penultimate buyer was eliminated. Typical
tv one
• Double Auction: K seller with 1 unit each to sell and many buyers. Sellers and buyers
simultaneously submit bids. The K highest bids get an object.
- The price is determined by an average of the K and K+1 highest bids.
- Sellers with bids above this price don’t sell their object, sellers with bids below this price
do sell.
• Single Unit Auction Model: one seller, N buyers who have a value vi∈[0,1] for the object
• If bidder wins at price p, utility= vi-p else 0 if lose
• The realisation of (v1, v2,..., vN) determines the seller’s demand curve.
• Independent Symmetric Private Values: vi is drawn independently from density f(v) on [0,1]
• Independent Private Values: vi is drawn independently from the density fi(v) on [0,1]
- i.e. player i and j have different distributions of values
• Correlated Private Values: The vector (v1, v2,..., vN) is drawn from a full support density on [0,1]N
- vi and vj are related
• Common Values: v1=v2=...= vN but players do not know what v is. They have
private information on it. Never have different values hence probability is 0
• We consider Independent Symmetric Private Values.
• Players only observe their own value and then bid.
• Player’s strategies are bidding functions that maps the player’s value to a bid bi(vi)
• We only consider equilibria where bids are strictly increasing functions of values
• See older notes for 2nd price auctions, similar ideas apply for English Auctions
, Designing Economic Systems
• A seller can build the project and the buyer can use it. If the buyer and seller agree on terms, the project
goes ahead i.e. price buyer pays to seller
• The project is ‘easy’ for the seller with probability 1/2 and low cost C=35 and the project is ‘hard’ and
expensive at C=60
• Buyers have low value V=40 and high value V=60, both with probability 1/2.
• Sellers don’t know buyers’ values and vice versa. Can’t force participation
• We plot the gains from trade
• If there was complete information, trade is efficient 3/4 of the time
• Each call has a probability of (1/2)(1/2)= 1/4 so expected gains from trade are 1/4(5) +
1/4(0) +1/4(5) + 1/4(30) =10
• Myserson Satterhwaite Theorem: there is no bargaining procedure under which the project
goes ahead even when it’s efficient to do so.
• We organise a system of trade where we get the buyer and seller to tell us their values
and then choose a price that shares the gains from trade equally
• If the seller announces he is a low cost then the project always goes ahead
- 1/2 the time at a profit of 2.50 (37.5-35), and half the time at profit of 15 (50-35), for an
average of 8.75
• If the seller is low cost but pretends to be high cost then the project gets built 1/2 of the
time at a profit for the seller of 62.50-35= 27.5
• So expected profit from lying is 1/2(27.50)=13.75
• The seller makes a higher expected profit from lying so the low-cost seller will lie. This scheme fails
• As long as values are private information, there is no mechanism that gets players to tell the truth and
builds the project whenever it should be built (no efficient way to organise trade)
• it is impossible to get the buyer and the seller to truthfully reveal their type here
• To get a high-cost seller to trade with a high-value buyer, we must promise them a price of at least 60 half
the time. At an efficient outcome he will end up building the project 1/2 the time
• If the seller actually had low costs, he could lie and pretend to be high cost to get at least 1⁄2(60-35)=12.5
• The real mechanism must give the seller expected profits of at least 12.5 when he is low-cost to ensure
he tells the truth.
• The seller expects to earn at least 12.5 at least 1⁄2 the time (when low cost) and 1⁄2 the time gets 0
• The seller’s expected profit is therefore 6.25. We must give at least this surplus from trade to the seller
• The buyer can pretend to have low value so he must have expected profit of at least 6.25, independent of
the mechanism. We must give both people incentives to tell the truth
• But the expected gains from trade were only 10 but we need 12.5 (6.25+6.25)
• It’s impossible to arrive at an efficient outcome using bargaining/negotiation when there’s private info
Mechanisms and the Revelation Principle
• To describe a problem, we say: there are individuals in the economy. Set of possible allocations of social states: x ∈X.
• Each individual has a utility function ui(x,θi), depends on social state chosen and on their private type/ knowledge θi
• Individual i’s type is selected from a set Θi by a known probability distribution e.g. before it was 1/2 low cost
• A mechanism describes how planner organises- receive info from agents as signals, aggregate then make decision
• A mechanism provides each individual in society a set of signals Si that they can send . It also announces a policy
rule γ which turns the signals into an allocation x
- e.g. message is how much you value bridge. Policy rule is whether to build and who pays for it
• Process: set up mechanism, individuals observe their utility and type. The messages sent depend on their type so we
denote this as si*(θi) . This is a game because the messages one individual sends depends on what others do.
• Once all the messages are sent the allocation gets decided.
• The equilibrium is denoted as γ(s1*(θ1),..., sn*(θn)) =x
• The policy rule maps types to a decision
• We can write this as Γ(θ1,..., θn) = x
- We say it is possible for the planner to implement the decision rule Γ(θ1,..., θn) = x
• e.g. decision rule is to oblige the buyer and seller to trade iff buyer is high value and seller is low, this is possible
• Revelation principle: if strategies si*(θi) are an equilibrium in dominant strategies, we only need agents to tell us their
type Si =Θi . Don’t need any other messages
