ECON 383 Applied Game Theory Tutorial
VCG Mechanism
1. Why VCG mechanism is awesome?
1.1 In the VCG mechanism, truth­telling is a dominant strategy. → All agents always report their true value
● Let .... be agent j’s reported value and v j be agent j’s true value
● Then … = v j . → We can replace … with v j .
● Implication: we can calculate social welfare from reported values, instead of true value.
● VCG mechanism maximize social welfare. In other words, the VCG prices are market clearing prices.
2. What is VCG mechanism?
Main idea: every agent pays equal to the external cost he or she creates. The external cost created by agent j is equal to the change in summation of all agents welfare except agent j if agent j is absent.
Therefore, agent j pays
P j = V a −j − V p −j P j = VCG price for agent j
V a −j = maximum social welfare (excluded agent j’s value) in the absence of agent j
V p −j = maximum social welfare (excluded agent j’s value) in the presence of agent j
Example: An auction with one object
● 5 buyers with values v 1 > v 2 > v 3 > v 4 > v 5
● Only one object Question: How much each buyer pay in the VCG mechanism?
Recall 1) In the VCG mechanism, all buyers report their true values.
2) Social welfare is maximized when the object goes to the buyer with the highest value.
Consider Buyer 1….
● In the absence of buyer 1, the social welfare is maximized when buyer 2 win the auction. Therefore,
Buyer 1 2 3
welfare ­
v2 0
4
5
0
0
V a −1 = v 2 + 0 + 0 + 0 = v 2 ● In the presence of buyer 1, the social welfare is maximized when buyer 1 win the auction. Therefore
Buyer 1 2 3
4
5
welfare v 1 0 0
0
0
V p −1 = 0 + 0 + 0 + 0 = 0 ● Thus, P 1 = v 2 − 0 = v 2
Buyer 2,3,4 and 5
● Buyer 1 win the auction either the absence or the presence of buyer 2 or 3 or 4 or 5.
● Thus, V a −j = V p −j = v 1 and P j = v 1 − v 1 = 0 for any j = 2,3,4,5.
Note
1) This is exactly like the second­price auction. (second price auction is a special case of VCG mechanism)
2) Intuition
● Buyer 1 pay = v 2 because the presence of buyer 1 make buyer 2 (the second highest buyer) loss the opportunity to win the auction → the external cost = v 2 .
● Buyer 2,3,4 and 5 pay nothing because their presence don’t change the outcome in which buyer 1 always win the auction.
Homework
1) Find the VCG prices in the case that there are two objects
2) Think about question 5 in the midterm exam. In your own words, give the intuition why in the VCG mechanism a voter who votes yes should pay only if there are exactly 4 other voters vote yes.
3) There’s at least one possible issue of VCG mechanism implementation. Discuss.