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

**1 ^{st}
price **

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

**2)
****For each player i, the set of strategies StrategySet_{i}_{ }= { b_{i}_{ }| b_{i} 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_{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 (denoted as bid* _{i }*below)

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 highest price (the 1 ^{st}
price) on the bid 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 a true **value
v_{i}_{ }in** his/her mind

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

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

b)
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)
**Note: **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.**

*****************************************************************************

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

**1 ^{st}
price **

**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.**

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

For each player *i*, there is a
true **value v_{i}_{
}in**
his/her mind.

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