**Version #1, for Homework #4A**

**2 ^{nd} price
sealed-bid auction with billionaires in the auction**

**1)
**** n bidders as players**: 1,2,…, n.

**2)
****Each player i submits
a natural number with his/her ID then automatically attached after the decimal
point as the bid. This ensures that no two players can happen to have
the same number as their bids and there is no tie in the result. In other
words, the set of strategies StrategySet_{i }is {
b_{
}| b is in the decimal format of k.i where k is a positive
integer and the unique ID i.} For example, for player 28 the set of
strategies is StrategySet_{28 }= {
b_{
}| b is in the decimal format of k.28 where k is a
positive integer.} In other word, player 28 can bid 1.28, 2.28, 3.28,
4.28 and so forth. **

3) **Rules of the auction**:

a)
Each player*
i* submit his/her bid (denoted as bid* _{i }*below) in a sealed envelope to the
auction. No one other than player

b)
After all the players submit their bids to the
auction, the auctioneer opens the envelopes and arranges them **as a list** in descending order of the
bids. The player on the top of the list (i.e. the one offering **the highest bid**) **gets the auctioned item** and need to **pay the second highest price on the bidding
list.**

4) **Strategy profile**: The strategy profile
of the auction outcome is simply the vector < bid* _{1}*, bid

5) **Pay-off functions**: For each player *i*, the pay-off function π* _{i }*(bid

For each player *i*, there is some subjective constant
value **v_{i}**

a)
If player *i*
**does get the auction item and pay no
more than the value v_{i }in his/her mind for the item**,
the player is very happy, which is represented by

b)
If player *i*
**does not get the auction item**,
his/her happiness level is neutral, which is represented by a **payoff of 0**.

c)
If player *i*
**does get the auction item but pay more
than the value v_{i }in his/her mind for the item**,
the player feels it is not worthwhile and is very upset, which is represented
by

**6) **Note:
**The formulation
in 1) and 2) above ensures that there is no tie since no two players can happen
to submit the same number as their bids. **

*********************************************************************Version
#2 as a separate variant for Homework #4B**

**2 ^{nd} price
sealed-bid auction with ordinary folk in the auction
**

**In this version, only
the playoff function for each player is slightly changed to better model how ordinary people may feel about the auction
result.**

5) **New pay-off functions**: For each player *i*, the pay-off function π* _{i }*(bid

Again, for each
player *i*, there is some subjective constant
value **v_{i}**

a)
If player *i*
**does get the auction item and pay no
more than v_{i }for the item**, the player is
happy and his/her payoff is

b)
If player *i*
**does not get the auction item**,
his/her happiness level is neutral, which is represented by a **payoff of 0**.

c)
If player *i*
**does get the auction item but pay more
than the value v_{i }in his/her mind for the item**,
the player feels it is not worthwhile and is very upset, which is represented
by