Pinhole camera model

Revenue equivalence is a concept in auction theory that states that given certain conditions, any auction mechanism that results in the same outcomes (i.e. allocates items to the same bidders) also has the same expected revenue.

Revenue Equivalence Theorem

An auction is a special case of a mechanism. In this case, the mechanism takes buyers' bids and decides an outcome of the auction: who gets the object and what are the transfers for each buyer. The set of outcomes can be denoted by

$\{(x,t)\in {\mathbb{ℝ}}_{+}^I\times {\mathbb{ℝ}}^I|x_i\in \{0,1\},\sum _{i=1}^I{x}_i=1\}.$

The x component describes the allocation of the object and t the transfers.

The buyer's types, or valuations of the object, are independent identically distributed random variables. A buyer of type ${\theta }_i$ has linear utility function u_i over the set of outcomes (the theorem also holds for the more general quasilinear utility functions):

$u_i(x,t,{\theta }_i)={\theta }_i{x}_i+t_i.$

Thus an auction is a Bayesian game in which a player's strategy is his bid as a function of his type. An auction (more generally, a mechanism) is said to be Bayesian incentive compatible if all players bidding their true type is a Bayesian Nash equilibrium strategy profile.

Under these assumptions, the Revenue Equivalence Theorem then says the following:

Theorem For any two Bayesian incentive compatible auctions, if under their respective Bayesian Nash equilibria where all players bid their type,

1. a buyer of type θ_i has the same probability of getting the object across auctions, and
2. a buyer of lowest type has the same expected utility across auctions,

then the total expected transfers E_θ(Σt_i), i.e. the auctioneer's expected revenue, is the same for the two auctions.

In other words, if a buyer of given type has the same expected utility in the two auctions in the interim stage, then the seller's expected revenues are the same. However, ex post, the two mechanisms need not implement the same social choice functions. Two such examples are the second price auction and first price auction. Assume the types are drawn independently from the uniform distribution on [0,1]. In the second price auction, bidding one's own type is a dominant strategy, therefore a fortiori the auction is Bayesian incentive compatible. For the first price auction, it can be shown that the bid functions

$b_i(\theta )=\frac{I-1}{I}({\theta }^{I-1})$

form a Bayesian Nash equilibrium (a simple argument via the revelation principle shows it can be made Bayesian incentive compatible). Thus the Revenue Equivalence Theorem applies: in both auctions, the highest types get the object and a buyer of type 0 has zero expected interim utility. They do not implement the same social choice functions.

Equivalence of Auction Mechanisms in Single Item Auctions

In fact, we can use revenue equivalence to prove that many types of auctions are revenue equivalent. For example, the first price auction, second price auction, and the all pay auction are all revenue equivalent.

Second Price Auction

Consider the second price single item auction, in which the player with the highest bid pays the second highest bid. It is optimal for each player $i$ to bid her own value $b_i=v_i$.

Suppose $i$ wins the auction, and pays the second highest bid, or ${\max }_{j\ne i}{b}_j$. The revenue from this auction is simply ${\max }_{j\ne i}{b}_j$.

First Price Auction

In the first price auction, where the player with the highest bid simply pays her bid, if all players bid using a bidding function $b(v)=E({\max }_{j\ne i}{v}_j|v_j\le v\mathrm{\forall }j)$, this is a Nash equilibrium.

In other words, if each player bids such that they bid the expected value of second highest bid, assuming that theirs was the highest, then no player has any incentive to deviate. If this were true, then it is easy to see that the expected revenue from this auction is also ${\max }_{j\ne i}{b}_j$ if $i$ wins the auction.

Proof

To prove this, suppose that a player 1 bids $b(z)<v$, effectively bluffing that her value is $z$ rather than $v$. We want to find a value of $z$ such that the player's expected payoff is maximized.

The probability of winning is then $Pr({\max }_{i>1}{v}_i<z)$. The expected cost of this bid is $E({\max }_{i>1}{v}_i|v_i<z\mathrm{\forall }i)$. Then a player's expected payoff is

$Pr({\max }_{i>1}{v}_i<z)(v-E({\max }_{i>1}{v}_i|v_i<z\mathrm{\forall }i))$

Let $X={\max }_{i>1}{v}_i$, a random variable. Then we can rewrite the above as

$Pr(X<z)(v-E(X|X\le z))$.

Using the general fact that $E(X|X\le z)\cdot Pr(X<z)=\underset{0}{\overset{\mathrm{\infty }}{\int }}Pr(X<z)-Pr(X<y)dy$, we can rewrite the above as

$Pr(X<z)\cdot v-Pr(X<z)\cdot z+\underset{0}{\overset{z}{\int }}Pr(X<y)dy$.

Taking derivatives with respect to $z$, we obtain

$Pr(X<z{)}^{\prime }(v-z)=0\Rightarrow v=z$.

Thus bidding with your value $v$ maximizes the player's expected payoff. Since $Pr(X<z)$ is monotone increasing, we verify that this is indeed a maximum point.

Using Revenue Equivalence to Predict Bidding Functions

We can use revenue equivalence to predict the bidding function of a player in a game. Consider the two player version of the second price auction and the first price auction, where each player's value is drawn uniformly from $[0,1]$.

Second Price Auction

The expected payment of the first player in the second price auction can be computed as follows:

$E(\text{Payment}|\text{Player 1 wins})P(\text{Player 1 wins})+E(\text{Payment}|\text{Player 1 loses})P(\text{Player 1 loses})$

Since players bid truthfully in a second price auction, we can replace all prices with players' values. If player 1 wins, he pays what player 2 bids, or $p_2=v_2$. Player 1 himself bids $p_1=v_1$. Since payment is zero when player 1 loses, the above is

$E(v_2|v_2<v_1)P(v_2<v_1)+0$

Since $v_1,v_2$ come from a uniform distribution, we can simplify this to

$\frac{v_1}{2}\cdot v_1=\frac{v_1^2}{2}$

First Price Auction

We can use revenue equivalence to generate the correct symmetric bidding function in the first price auction. Suppose that in the first price auction, each player has the bidding function $b(v)$, where this function is unknown at this point.

The expected payment of player 1 in this game is then

$E(\text{Payment}|\text{Player 1 wins})P(\text{Player 1 wins})+E(\text{Payment}|\text{Player 1 loses})P(\text{Player 1 loses})$ (as above)

Now, a player simply pays what the player bids, and let's assume that players with higher values still win, so that the probability of winning is simply a player's value, as in the second price auction. We will later show that this assumption was correct. Again, a player pays nothing if he loses the auction. We then obtain

$b(v_1)\cdot v_1+0$

and by revenue equivalence,

$b(v_1)\cdot v_1=\frac{v_1^2}{2}\Rightarrow b(v_1)=\frac{v_1}{2}$

Indeed, with this bidding function, the player with the higher value still wins. We can also show that this is the correct equilibrium bidding function, by thinking about how a player should maximize his bid given that all other players are bidding using this bidding function.

All-pay Auctions

Similarly, we know that the expected payment of player 1 in the second price auction is $\frac{v_1^2}{2}$, and this must be equal to the expected payment in the all-pay auction, i.e.

$\frac{v_1^2}{2}=b(v_1)$

Thus, the bidding function for each player in the all-pay auction is $\frac{v^2}{2}$

Further reading

• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.