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

1) *n *bidders as players: 1,2,…, n

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}‘s are
disjoint, all bids are distinct, and there is no tie just like in Homework 4A
and Homework 4B**. Note that the reward for the winner more precisely should
be

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 is the set of
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 reached some conclusions as his/her knowledge base for further reasoning and also assumes other rational players have reached the same conclusions as the knowledge base for further reasoning. If the players then repeat the 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.)