**Guess-½-of-Average
Sealed-Bid Lottery (Version #1 for Homework #5)**

**1)
***n *bidders
as players: 1,2,…, n for **a large n (say n > 10000)**

2) For
each player *i*,
the set of strategies **StrategySet_{i}**

3) Rules of the lottery:

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 calculates ½ of the average of
all the submitted numbers. **The winner is
the player whose number b_{i
}_{ }is closest
to the ½ of the average**

c)
**Note
about ties**: To make it simple, the auctioneer attaches **arbitrarily** small distinct epsilons (as fixed distinct IDs
associated with the payers as described in Homework 4A and Homework 4B) to the bids
of the players. In other words, all strategy sets **StrategySet_{i}‘**

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

5) The
happiness payoffs: For each player *i*, the **payoff is
simply the amount of money rewarded by the lottery.**

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

**Questions**:

1. For
a player *i*,
does 127 (as a strategy) weakly dominate 128?
If so, prove it. If not, give a scenario of b-_{i}* _{ }*in which the payoff of the
strategy profile (b

2. For
a player *i*,
does 126 (as a strategy) weakly dominate 127?
If so, prove it. If not, give a scenario in which the payoff of the
strategy profile (b* _{i}*=127,
b-

3. For
a player *i*,
does 64 (as a strategy) weakly dominate 65?
If so, prove it. If not, give a scenario in which the payoff of the
strategy profile (b* _{i}*=65,
b-

4. For
a player *i*,
does 63 (as a strategy) weakly dominate 64?
If so, prove it. If not, give a scenario in which the payoff of the strategy
profile (b* _{i}*=64, b-

5. For
a player *i*,
does 62 (as a strategy) weakly dominate 63?
If so, prove it. If not, give a scenario in which the payoff of the
strategy profile (b* _{i}*=63,
b-

6. For
each player *i*,
does 1 (as a strategy) weakly dominate all other strategies? If so, prove it.
If not, give a scenario in which the payoff of the strategy profile (b* _{i}*=1, b-

7. For
each player *i*,
is there a strategy that weakly dominates all other strategies? If so, prove
it. If not, explain why not.

8. For
each player *i*,
what are the weakly dominated strategies (i.e. each of them is weakly dominated
by at least one other strategy)? Are there reasons that a rational player may
want to use any of these weakly dominated strategies?

9. **After the reasoning above, a rational
player has an understanding about dominated strategies as “bad ideas” to avoid
and may assume other rational players have the same understanding. If the
players then repeat a similar process of reasoning from #1 to #8 again and
again, what eventually would their strategy (strategies) be for the lottery**?

10. Think
about #9 above again in the context of #1 and #2 in the summary of chapter 4 of *Strategies and Games: Theory and Practice.* Is there an** IEDS** solution
for the *Guess-½-of-Average
Sealed-Bid Lottery*? If so, what is the **IEDS** solution?

**Guess-½-of-Average
Sealed-Bid Lottery (Version #2)**

**In this version, the award for the winner in the lottery game is
changed from (128 – b_{i}) to
b_{i}
instead. **

What would happen to your answers
above? (No need to answer them in the Homework just do it as an exercise on
your own.)

[1]
To ensure the reward is non-negative, the reward for the winner actually should
be **(maximum of the epsilons) + 128 - b_{i }_{ }**depending on the