9. Models of Value and Power
Your value will be not what you know; it will be what you share.
—Ginni Rometty
In this chapter, we cover models that quantify the value and power of
individual actors. Some cases are easy. When a group produces output
equal to the sum of individual contributions, each individual’s value equals
her contribution. When the collective output cannot be separated into
individual components, such as when a team of computer programmers
writes a software program or a group of entrepreneurs proposes creative
uses for a new technology, assigning credit becomes difficult. Assigning
power to political parties creates similar problems; the number of seats a
party controls correlates with power, but not perfectly.
In this chapter, we define two measures of value and power: last-on-the-bus
value, which equals an actor’s marginal contribution given that the group
has already formed, and Shapley value, which equals an actor’s average
marginal contribution across all possible sequences of adding people to a
group. In a group of three people, we average a person’s added value when
she joins the group first, second, and third. We define these measures
within the structure of cooperative game models, which consist of a set of
players along with a value function that assigns a collective payoff to every
possible subset of the players.
The chapter consists of four parts. In the first part, we define cooperative
game models, last-on-the-bus value, and Shapley value, and work through
some examples. In the second part, we describe axiomatic foundations for
the Shapley value. We show it to be the unique measure satisfying four
conditions. One condition is that a player who never adds value must be
assigned value zero. A second condition is that the sum of the player’s
values must equal the total value of the game. In the third part, we apply
Shapley value to a group performing a creative task. Each person thinks up
ideas. We show how in this context, the measure produces an intuitive
measure of value. In the fourth part, we consider the special case of
applying the Shapley value to voting games. We use it to distinguish
between voting power and vote percentage. We find that they need not
always agree. A party might hold 20% of the seats and have no power in
one case and a third of the total power in another.
Cooperative Games
A cooperative game consists of a set of players and a value function that
assigns a value to every possible subset, often called a coalition, of players.
Cooperative games are meant to capture collective work and joint projects.
In the model, we assume that people participate so that we can focus
attention on how to assign value to their participation.
Cooperative Games
A cooperative game consists of a set of N players and a value function
that assigns a value to any subset S ⊆ N, V(S). These subsets are called
coalitions. The value of the coalition consisting of no players equals zero,
image = 0; the value of all N players, V(N), equals the total value of the
game.
In a cooperative game, a player’s last-on-the-bus (LOTB) value equals the
value she adds if she is the last to join the group. LOTB values capture
players’ values at the margin. If four people are hired to move a table, and
moving the table produces a value of 10, and all four are needed, then each
has a LOTB value of 10. If only three are required, then each has a LOTB
value of zero. Notice that LOTB values need not sum to the total value of
the game. In particular, if the value function exhibits diminishing returns to
scale, then LOTB values sum to less than the total value, and if added
values exhibit increasing returns to scale, then the sum of LOTB values
exceeds the total value.
A player’s Shapley value equals her marginal contribution when she is
added to a coalition averaged across all possible orderings in which the
coalition of everyone forms. In other words, we imagine adding the players
to the coalition in sequence and calculating a player’s added value for each
sequence. Consider a small firm that operates in Spain and France and
requires one French speaker and one Spanish speaker to conduct daily
business. The firm has three employees: a Spanish speaker, a French
speaker, and a bilingual person capable of speaking both French and
Spanish.
Suppose that our cooperative game assigns a value of $1,200 to any set of
workers capable of speaking French and Spanish. This amount equals the
daily revenue of the firm if it is able to operate. If any two employees show
up at the office, the third is not needed. Therefore, each player has an
LOTB value of zero.
To compute the Shapley value for the French speaker, we consider all six
orderings in which people could arrive to work. In only one of these
orderings, the one in which the Spanish speaker arrives first and the French
speaker arrives second, does the French speaker add value. Her Shapley
value equals image times $1,200, or $200. The Spanish speaker adds
value only if he arrives second and the French speaker arrives first, so his
Shapley value also equals $200. In the other four orderings, the bilingual
person arrives either first or second and adds value. Her Shapley value
therefore equals $800. The sum of the Shapley values equals $1,200, the
total value of the game.
Shapley Value
Given a cooperative game {N, V}, the Shapley value is defined as follows:
let O represent all N! orderings in which the N players could arrive and be
added to a group. For each ordering in O, define the added value of player
i to be the change in the value function that occurs when player i is added.
Player i’s Shapley value equals the average of her added values over all
orderings in O.
Now that we have the idea, we construct a more complicated example.
Imagine a crew team that requires four rowers and a coxswain—a smaller
person who manages the stroke rate and steers. Our crew team (the players
in the cooperative game) consists of six individuals: five tall, powerful
rowers and a smaller person who has been trained as a coxswain. To enter a
race, the team needs four rowers and a coxswain. A team of five that
includes the smaller, trained coxswain will be competitive and has value
10. A team of five rowers without the coxswain could enter a race, but
would perform poorly because of the extra weight. We assign that team of
five a value of 2.
To compute Shapley values, we imagine the players arriving in every
possible order. If the smaller coxswain arrives first, second, third, or fourth,
she adds no value. If she arrives fifth, which occurs one-sixth of the time,
she adds value 10. If she arrives sixth, she replaces one of the rowers as
coxswain and her added value equals 8. Averaging across all of these
possibilities, we find the coxswain’s Shapley value equals 3.
Each rower adds value if and only if she arrives fifth, which occurs onesixth of the time. If the coxswain has not arrived, the rower who arrives
fifth adds value 2. If the coxswain has arrived, the rower adds value 10.
Given the one-in-five chance the coxswain is last among the five other
players, and the four-in-five chance the coxswain arrives among the first
four, we arrive at a Shapley value of image for each rower.1 Intuitively,
the coxswain’s value should be more than the value of a single rower and,
given that the rowers can compete, albeit poorly, without the coxswain, less
than the combined value of all of the rowers. There are an infinite number
of ways to assign values that satisfy those constraints. Shapley values
assign specific values: 3 for the coxswain and 7 total for the five rowers.
Axiomatic Basis for the Shapley Value
We now describe a set of axioms that Shapley values uniquely satisfy. That
result explains why we might privilege Shapley values over other possible
measures. First, note that we calculate Shapley values by averaging a
player’s marginal contribution across all possible orderings, so any player
who never adds value has a Shapley value of zero. Moreover, any two
identical players—that is, two players who, for each coalition, contribute
the same amount—must be assigned the same Shapley value. And given
that the sum of the added values equals the total value of the game for any
ordering, Shapley values must also sum to the value of the game. These
will be three of the four axioms. Notice that LOTB values satisfy the first
two axioms but not the third.
To these three properties, we add a fourth, additivity, which requires that if
the value function of a cooperative game can be decomposed into two value
functions, each assigned to a different cooperative game, a person’s value
in the combined game should equal the sum of her values in the two
constituent games. A moment’s reflection reveals that Shapley value
satisfies this property as well. That those four properties uniquely
characterize the Shapley value is less obvious.
Showing that a measure uniquely satisfies a set of axioms places the
measure on logical foundations. Without the axioms, a measure may be
intuitive but could be seen as arbitrary, as one of several plausible
measures. The theorem also tells us that if we choose any other measure,
we must abandon one of the axioms. This does not mean that Shapley value
is the only reasonable measure. Lloyd Shapley, an economist and
mathematician, may have first written down the measure and only after the
fact constructed axioms that it uniquely satisfies. Which came first is of
little relevance. Even if the axioms had been backward-engineered, if we
accept the axioms, we should embrace the measure. The appropriateness of
the measure hinges on the reasonableness of the axioms. In this case, the
first three are difficult to dispute. The fourth, additivity, though more
complicated than the others, can be supported on the grounds that if it did
not hold, players would have incentives to split up or form coalitions.
Shapley Value: Axiomatic Basis
The Shapley value uniquely satisfies the following axioms:
Zero property: If a player’s added value equals zero for any coalition, the
player’s value equals zero.
Fairness/Symmetry: If two players have the same added value for any
coalition, then those players have the same value.
Full allocation: The sum of the values of the players equals the total value
of the game, V(N).
Additivity: Given two games defined over the same set of players with the
value functions V and image, the value of a player in the game (V +
image) equals the sum of that player’s values in V and image.
Shapley Values and the Alternative Uses Test
We now apply Shapley values to a cooperative game based on the
alternative uses test. In the test, each person must come up with novel uses
of a common object, such as a brick. The test measures a person’s creativity
based on the number of uses or categories of uses that she generates. When
we calculate Shapley values, we find that they produce an intuitive scoring
rule.
Imagine three players, Arun, Betty, and Carlos, who each think up
alternative uses for blockchain, a distributed ledger technology, shown in
figure 9.1. Arun and Carlos each think of six ideas, giving each a creativity
score of 6, and Betty thinks of seven, making her score 7. The group’s total
creativity equals 9, as there are nine unique ideas. To compute the Shapley
values, we could write down all six possible orders in which the group
could form, give individuals credit only for unique ideas added to the
group, and then average over all six cases. Or we can notice that when we
are computing Shapley values, the probability of someone getting credit for
an idea equals 1 divided by the number of people who propose the idea.
Anyone who proposes a unique idea always receives full credit. In the
figure, we denote those ideas, such as Arun’s idea of art transactions, in
bold font. If two people think of an idea, each has a one-in-two chance of
joining the group first. Similarly, if all three people think of an idea, each
has a one-in-three chance of joining first. It follows that allocating credit
equally among people who thought of an idea produces Shapley values.
Thus, it is the unique way to assign values that satisfies the four axioms.
These values show that Arun, though he did not have the most ideas, adds
the most value.2
image
Figure 9.1: Shapley Values and the Alternative Uses Test
The Shapley-Shubik Index
We next apply Shapley values to a class of voting games. In a voting game,
each player (representing a political party or official) controls a fixed
number of seats or votes, and a majority of those seats or votes are
necessary for taking an action. In voting games, the Shapley value is
referred to as the Shapley-Shubik index of power.3 By calculating the index,
we find there does not exist a direct translation between the percentage of
seats (votes) a party controls and its power.
To compute power indices, we consider all possible orderings of parties
being added to a coalition. If a party joins the coalition and creates a strict
majority, the party’s added value equals 1. In those cases, the party is said
to be pivotal. Otherwise, the party adds no value. Consider a parliament
with 101 seats allocated across four political parties as follows: party A
controls 40 seats, party B controls 39 seats, and parties C and D each
control 11 seats. In this example, party A cannot be pivotal if it arrives first
or last. If party A arrives second or third, it is always pivotal. Therefore, it
has a power index of image. If party B arrives first or last, it also adds no
value. If B arrives second, it is pivotal only if party A arrived first. If party
B arrives third, the only way that it can be pivotal is if party A arrives last.
Each of those combinations of events also occurs with probability image.
Therefore, party B’s power index equals image. Parties C and D are
pivotal in a similar set of cases as party B. Neither can be pivotal if it
arrives first. Each is pivotal if it arrives second only if party A arrived first.
Each is also pivotal if it arrives third when party A arrives last. Thus, each
of those parties also has a power index of image.
image
Figure 9.2: The Disconnect Between Seats and Power
The example reveals a possible disconnect between the percentage of seats
a party controls and its power. Parties A and B control almost identical
numbers of seats, but A has three times the power of party B, which has no
more power than party C or party D. Similar allocations of seats occur
often in real-world parliamentary systems. As a result, parties with few
seats can often control substantial power. Israel’s parliament, the Knesset,
has 120 seats. In 2014, a coalition led by the Likud party had 43 seats. The
opposition coalition had 59 seats (just shy of a majority), and an Orthodox
coalition held eighteen seats. All three parties have the same ShapleyShubik index. This does not mean that the small Orthodox parties have the
same power in practice—all models are wrong. It does suggest that the
Orthodox parties had more influence than would be anticipated from their
seat count.
An even more stunning disconnect between seats and power occurred with
the Nassau County Board of Supervisors in New York in the mid-1960s. At
that time, the board consisted of six members, and each controlled votes
proportional to the population of the districts she represented, as shown in
figure 9.3. A majority vote required 58 or more of the 115 votes. Notice
that any two of the three largest districts constituted a majority. It follows
that the votes of the other three districts could never be pivotal. The other
district representatives therefore had no power. The North Hempstead
representative controlled 21 votes, more than 18% of the total yet could not
influence a voting outcome.
image
Figure 9.3: Votes but No Power
The Shapley-Shubik index of power can be applied to any situation with
unequal allocations of seats or votes, such as the European Union or the
Electoral College. That does not mean that it is necessarily an appropriate
measure in all cases. The fifty states can be arranged in 50! (3 × 1064)
different orders. Given regional correlations in voter preferences, not all
coalitions are possible. Mississippi may not be likely to form a coalition
with New York. To make a more useful measure of power we would need
to privilege some coalitions over others or rule out some coalitions
altogether. Later in the book, we describe Myerson values, which allow us
to do the latter, to rule out some coalitions.
Summary
An individual’s Shapley value corresponds to her average added
contribution to coalitions as they form. It is a measure of added value. In
voting games, Shapley value can also be interpreted as a measure of power.
It may not always be the best measure of power. An individual’s LOTB
value may be the better measure of power in situations where a group has
already formed, as that measures how much each individual could extract
through a threat to leave, assuming that threat is credible.
In those cases, the coalition wants to reduce LOTB values. Creating a
coalition with a high value but low LOTB values can be accomplished by
increasing the coalition size. Adding extras makes existing members
expendable and drives their LOTB values to zero. We see this in practice.
Employers hire excess workers to reduce worker power. Manufacturing
firms rely on multiple competing suppliers of intermediate goods.
Governments award contracts to keep multiple contractors in business.
The same intuition applies to the creation of coalitions in legislatures.
Congressional lobbyists and party leaders want to pass legislation (an
outcome of value) but restrict the power of individual representatives and
senators.4 If a lobbyist makes contributions to the minimal number of
representatives and senators necessary to win a vote, then each
representative and senator has an enormous LOTB value. Any could switch
his or her vote and flip the outcome of the bill. The lobbyist can reduce
their LOTB values by buying a supermajority of representatives and
senators. The same logic implies that a party that holds a slim majority may
be difficult to lead. Every member has a large LOTB value. Within a strong
majority, no representative or senator has much power.
If we broaden our perspective and contemplate power in the modern
connected world, we find it useful to apply both LOTB values and Shapley
values. The power of an individual, organization, corporation, government,
or terrorist group depends partly on how much damage it could do by
deviating from a cooperative regime (LOTB value). A sophisticated
computer hacker, a person capable of destroying a substantial amount of
wealth, has enormous power. This holds even though the hacker lacks the
ability to add value.
In thinking about the value of corporations or other multinational
organizations, Shapley value may be a better measure. In these cases, exit
may not be a viable option. An energy company participates in an energy
generation game, an energy distribution game, a real estate game, an
environmental game, an employment game, and so on. The company’s total
added value equals the sum of its added values across the various domains.
Thinking of power and value through the lens of cooperative game theory
provides powerful, basic insights. It also points to where we should look
next. In politics and business, not all coalitions are plausible. The model
assumes that they are. A richer model would take into account the
connectedness of the world. Consulting companies and financial firms buy
software from tech companies. Tech companies and consulting companies
invest and borrow through financial firms. And financial firms and tech
companies hire consultants. Within those webs, each actor adds value and
wields power. To calculate power in these settings requires models of
networks, where we turn next.