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P12AMITM@iimidr.ac.in | LogoutGame Theory II 0

1. Part 1. VCG Variants

Suppose we wish to allocate an item to a set of 10 agents. For every integer , exactly one agent values the item at $v. (So one agent values the item

at $1, one agent values the item at $2, and so on up to $10). The outcome is thus which agent is allocated the item. Consider the following mechanisms:

X: The outcome and payments are chosen using VCG.

Y: The outcome and payments are chosen using VCG, and then each agent receives 1/10 of the total payments.

Z: Each agent pays a $2 fee, and then the outcome and payments are chosen using VCG.

For which mechanisms is truthtelling a dominant strategy?

X

Y

Z

X and Y

X and Z

Y and Z

All three mechanisms

None of the mechanisms

2. Which mechanisms are Groves mechanisms?

X

Y

Z

X and Y

X and Z

Y and Z

All three mechanisms

None of the mechanisms

3. Which mechanisms are weakly budget balanced in this environment?

X

Y

Z

X and Y

X and Z

Y and Z

All three mechanisms

None of the mechanisms

4. Which mechanisms are ex post individual rational in this environment?

X

Y

ZX and Y

X and Z

Y and Z

All three mechanisms

None of the mechanisms

5. Which mechanisms are budget balanced in this environment?

X

Y

Z

X and Y

X and Z

Y and Z

All three mechanisms

None of the mechanisms

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6. Part 2: Single-item VCG

Suppose that we wish to use VCG to allocate a block of bandwidth to one of three companies: X, Y , and Z. X and Y are competitors in the same industry; Z is in

a different industry. Their payofor the three possible allocations are:

Agent U(Allocate to X) U(Allocate to Y) U(Allocate to Z)

X 10 -15 0

Y -12 5 0

Z 0 0 4

To which company will VCG allocate the block?

X

Y

Z

7. What will be the winning company's payment?

0

1

2

5

-1

-2

-5

8. Does this environment exhibit choice set monotonicity?

Yes

No

9. Does this environment exhibit no negative externalities?

Yes

No

10. Does this environment exhibit no single-agent effect?

Yes

No

11. Part 3: VCG with Incomplete Reports

Suppose that we are using VCG to allocate a pair of shoes among three agents.Each agent can receive one shoe, both shoes, or neither shoe. The feasible

outcomes are those where each shoe is allocated to at most one agent.

The agents value the shoes as follows:

Agent U({left}) U({right}) U({left, right})

X 2 4 6

Y 0 0 7

Z 4 2 6

However, instead of reporting completely, the agents report only their utility for getting both shoes, as follows:

Agent U`({left}) U`({right}) U`({left, right})

X 0 0 6

Y 0 0 7

Z 0 0 6

Are the agents playing a dominant st rategy?

Yes

No

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12. Are the agents playing a Nash equilibrium?

Yes

No

13. What allocat ion will VCG choose given the reported preferences?

Both shoes to X

Both shoes to Y

Both shoes to Z

Left shoe to X, right shoe to Y

Left shoe to X, right shoe to ZLeft shoe to Y , right shoe to Z

Right shoe to X, left shoe to Y

Right shoe to X, left shoe to Z

Right shoe to Y , left shoe to Z

14. What is the efficient allocation given the true preferences?

Both shoes to X

Both shoes to Y

Both shoes to Z

Left shoe to X, right shoe to Y

Left shoe to X, right shoe to Z

Left shoe to Y , right shoe to ZRight shoe to X, left shoe to Y

Right shoe to X, left shoe to Z

Right shoe to Y , left shoe to Z

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