**2 ^{nd} price
sealed-bid auction (Version #1, for Homework #4A)**

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

**2)
****For each player i,
the set of strategies StrategySet_{i }=
{ b_{i }| b_{i} as 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_{28 }| b_{28} is in the decimal format of k.28 where k is a
positive integer.} In other words, player 28 can only bid 1.28, 2.28, 3.28,
4.28 and so forth. This ensures that all the sets StrategySet_{i}‘s are disjoint and no two players can
happen to submit the same number as their bids. **

3) **Rules of the auction**:

a)
Each player*
i* submit his/her bid b* _{i }*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 price on the bid list.**

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

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

For each player *i*,
there is a true **value v_{i}**

a)
If player *i*
**does get the auction item and pay no more
than the a true value v_{i }for the item**, the player is so
excited and his/her happiness level is very high, 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 a true value v_{i }for the item**, the player is
very upset and his/her happiness level is very low, which is represented by

**6)
****The formulation in 1), 2), and 5) above ensures that there is
no tie since no two players can happen to submit the same number as their bids.
In addition, player i is the only
person who can possibly bid the number v_{i},
the true value v_{i }for player i.**

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**2 ^{nd} price
sealed-bid auction (Version #2 as a separate variant for Homework #4B)**

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

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

For each player *i*, there is a true **value v_{i}**

a)
If player *i*
**does get the auction item and pay no
more than the a true value 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 a true value v_{i }for the item**, the player is
very upset and his/her happiness level is very low, which is represented by