Oct 10, 2007 - The judgment of king Solomon is an early example of mechanism design. Mechanism design attempts to achieve desired outcomes in ...
Examples of Mechanism Design: from King Solomon to eBay Dr. Mathijs de Weerdt October 10, 2007
The judgment of king Solomon is an early example of mechanism design. Mechanism design attempts to achieve desired outcomes in situations with self-interested players by setting the rules of the game in a speci c way. We will see that the game Solomon proposed would not have worked if the women were rational, and we will see an alternative mechanism (or game) as proposed by Moore (1992). Then we will relate this mechanism to another mechanism, namely the auction protocol used in e-Bay. Finally, we brie y introduce some mechanism design problems we study in the Algorithmics group (under prof. dr. Cees Witteveen).
The judgment of king Solomon Let us look at the story of the judgment of king Solomon from the Old Testament, First Book of Kings, chapter 3. At some point two women come to king Solomon with a child, because they are ghting over which one of them is the real mother of this child. One of them claims the other has swapped the children overnight because hers has died. Solomon listens to their story and then orders a soldier to cut the child in half, giving a half to each of the women.
Upon this one of the women then
cries: If it please you, my lord, let them give her the child; only do not let them think of killing it!. King Solomon then decides to let the child live, and gives it to this woman, for she must be his real mother. King Solomon assumed that the real mother would rather have the child living with the other woman, and that the false mother would prefer both children dead over being the only one who lost her child. By his little game he tried to retrieve the preferences of both women, from which he then could derive which woman was the real mother. Let us look at this situation more closely, following an analysis by Moore (1992). Let us call the rst mother Alice, and the second mother Barbra. Then we assign letters to each of the possible outcomes: outcome outcome
c
a
is to give the child to Alice, outcome
is to cut the child in half.
the real mother, and
β
is to give the child to Barbra, and
α
where Alice is
where Barba is the real mother. Solomon's assumption about the women's
preferences can then be summarized as follows: in state
a b c,
b
We know there are two possible states:
while Barbra prefers
b c a.
In state
β,
α,
Alice has the following preferences:
the roles are reversed and the Alice prefers
Figure 1: The Judgment of Solomon, 1518-19, Fresco, Loggia on the second oor, Palazzi Ponti ci, Vatican
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a c b,
while Barbra prefers
b a c.
This reversal is exactly what distinguishes one state
from another. Solomon determines the outcome, so in fact he de nes a choice function that maps a state of nature ∈
f (α) = a
{α, β} to an outcome ∈ {a, b, c}. and f (β) = b.
His goal is to de ne the following choice function
f:
However, he does not know the current state. So his problem is to de ne a mechanism or game
g
with the property that if
state
β
the outcome is
b.
g
is played in state
α
the outcome is
a,
and that if
g
is played in
In such a game each of the players (in this case, women) can choose a
strategy, and depending on both strategies the outcome is selected. In Solomon's game the women can choose between announcing give the child to her or I keep the child. Such games can be represented in a matrix. Each row
i
represents a strategy for Alice and each column
a strategy for Barbra. Given a strategy
(i, j).
i
and a strategy
j,
j
represents
the outcome can be read from the cell
In the case of Solomon, the matrix game looks as follows. give to rst mother
keep
give to second mother
?
keep
b
a c
Note that Solomon would not have known who the real mother was if both women had acted the same. He knew, however, that the real mother would prefer to give the child to the other mother to prevent him from killing it. So he would not get to outcome
c
if the real mother was rational.
However, had the other woman thought for a moment, she could have known that Solomon was trying to distinguish the real mother. If she were rational too, she would have acted similar to the real mother, i.e., saying give the child to her. There appears to be no way to adapt Solomon's mechanism in a simple way to prevent such behavior.
Alternative mechanisms There are, however, other mechanisms that could have been chosen by king Solomon and that would have worked for rational women as well. Suppose that everyone (both mothers and Solomon) knows that the real mother cherishes raising the child more than the false mother does. This value the mothers assign to raising the child is called the private value of each of the mothers. If Solomon introduced a mechanism in which he could nd out which woman has a higher private value for the child, he would know who the real mother is. Let us look at such a mechanism in Figure 2 where he levies a ne
F > 0 that is smaller than both women's valuations of the child,
if the women keep
on arguing (Moore, 1992). It is important that king Solomon rst explains the mechanism from Figure 2 to both women, and that the women have enough time to reason about what can happen in both situations (when Alice is the real mother, and when Barbra is the real mother). To see what the women should do, it works best to reason backwards from step 3. Try to see for yourself what you would do if you were Alice, but you would not be the real mother (so if your value for the child were smaller than Barbra's value). You will see that this depends on the value you would announce a value
v
v.
So if you were Barbra (in step 2),
as high as possible (but not higher than your value for the child).
Knowing this, it is best for Alice in step 1 to give up at once! A similar reasoning holds when Alice is the real mother. Alternatively, Solomon could have used an auction (see for example the discussion about the Vickrey auction in the next section). However, he did not want to receive money from the women. His only goal was to get to know which of the women had a higher valuation.
The Vickrey auction The idea of trying to get the private valuations of the players and use those to determine the optimal social choice is also what is used in eBay. At rst sight, eBay uses an open-cry English auction. However, when you submit a bid in eBay, you can choose to submit the real maximum amount you are prepared to pay. If another bidder bids below your maximum, your current bid is automatically increased to one increment above the bid of the other. This maximum bid is kept hidden from the other bidders.
If we view the auction protocol based on just these maximum
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1.
Alice: your child?
no
give child to Barbra
yes 2. Barbra: your child?
no
give child to Alice
yes Alice pays ne F Barbra announces value v > F
3. Alice: your child?
no
give child to Barbra Barbra pays ne v
yes give child to Alice Alice pays ne v Barbra pays ne F Figure 2: This mechanism solves Solomon's problem if the women are rational.
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Brief biography of William Vickrey At October 8th 1996 William Vickrey was awarded the Nobel prize in economic sciences. He died of a heart attack just three days later. Vickrey had many original ideas, combining mathematics and economics, and he usually tried to take these ideas all the way to the eventual use in practice. All his life he went from one real-life project in industry to another, selecting them mostly for the potential of improving human welfare (Lowell Harriss, 2000). For example, in 1959 Vickrey studied road transportation in Washington, D.C. By calculating the social marginal costs of each additional car on the road, he came to a highly dierentiated road tari structure according to the expected tra c congestion. Since he did not like the idea of tollbooths, he proposed a system where small radio transmitters would transmit vehicle or driver IDs to bill drivers automatically. In the mid-1960's, Vickrey was challenged that such a system was infeasible. He then responded in a typical fashion: he built a rudimentary computer in his home and connected it to a radio receiver, and built a small radio transmitter (for under $3) placed under the hood of his car. He could then show a printout of the times his own car went up or down his driveway to anyone who asked. From one of the printouts, it showed that as someone who practiced the concern for e ciency in transportation that he preached, Vickrey rarely used his car: he almost always took the train into Manhattan, then commuted the blocks from the station and across Columbia's campus to his o ce on roller skates (Harstad, 2005).
Figure 3: William Vickrey (1914-1996)
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bids, ignoring all current bids, eBay actually uses a sealed-bid second-price auction: the bidder that submitted the highest maximum gets the item for the price of the second-highest maximum (rounded to the next increment). This type of auction has the property that each of the bidders is best o submitting his private value as his maximum amount. A mechanism that has this property is called truthful or incentive compatible : assuming that everyone is rational, players cannot be better o by lying about their valuations. The sealed-bid second-price auction is called a Vickrey auction after its inventor (Vickrey, 1961) (see Figure 3 for a brief biography of Nobel prize winner Vickrey).
Research in the Algorithmics group In our research we study mechanisms such as the Vickrey auction (e.g., VCG-like mechanisms (Vickrey, 1961; Clarke, 1971; Groves, 1973)) to deal with coordination problems among self-interested agents. For example: how should the gates at Schiphol airport be scheduled, considering the preferences of all the airlines? Or: which transportation orders should be given to which transport company?
References Clarke, E. H. (1971). Multipart pricing of public goods. Public Choice, 11(1):1733. Groves, T. (1973). Incentives in teams. Econometrica, 41(4):61731. Harstad, R. M. (2005).
William S. Vickrey.
Working Papers 0519, Department of Economics,
University of Missouri. Lowell Harriss, C. (2000). William Spencer Vickrey, 1914-1996 Nobel laureate in economics. The Economic Journal, 110(467):F708F719.
Moore, J. (1992).
Implementation, contracts and renegotiation in environments with complete
information. In Laont, J. J., editor, Advances in Economic Theory, Volume 1, pages 182282. Cambridge University Press. Vickrey, W. (1961). Computer speculation, auctions, and competitive sealed tenders. Journal of Finance, 16:837.
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