MANAGEMENT TIP
Sharing the North Fork Canyon Run
Insights from Game Theory
for Cooperatives
By Bruce J. Reynolds, Ag Economist
USDA Rural Development/Co-op Programs
bruce.reynolds@usda.gov
t is puzzling that many businesses and
individuals forgo opportunities to join or
organize cooperatives when such actions
would benefit them. One reason for such
missed opportunities is that “go-it-alone”
decisions sometimes offer more immediate payoffs, or more
certainty of outcome, than do efforts that involve sharing
resources or participating in orderly marketing efforts.
Game Theory analysis helps identify situations that may
lead to coordinated decisions among various “players,”
depending on the way incentives are structured. An incentive
structure can be conveniently displayed in a 2 x 2 pay-off
matrix. These matrices provide a way to distinguish between
“dominated” and “contingent” choices, which is a key to
understanding the prospects for coordinated decisions. This
analysis will be applied in this article to hypothetical rafting
businesses that share a thin strip of white water on the North
Fork Canyon Run.
Game Theory is applied to studying situations where payoffs
to each participant are interdependent, i.e., determined
not only by the decision of an individual but also by the
decisions of others. For the sake of simplicity, this will be
examined as a two-person game.
Although the pay-offs are interdependent — mutually
affected by how many rafts in total are operated — the
decisions about how many rafts to operate are often
independent of what the other participant does. In these
cases, the incentive structure is based on dominated choices
— that is, the decisions of one participant do not influence
the other operator’s decision and vice versa.
Two-person game
The North Fork Canyon Run is a narrow branch of a river
that runs through a large and mountainous park. Several
small rafting companies provide tourists with white water
rafting recreation during the non-winter months. These trips
primarily are made on a couple of the larger rivers in the
park and, to a limited extent, at the North Fork Canyon Run.
This run is navigable only during a couple months in the
spring. Due to its narrow channels and frequent spots where
rafts get briefly hung-up, the park authority established a
one-raft-at-a-time rule. The park provides a dedicated phone
line between the entry and exit points so the rafters know
when to start the next raft trip. Rafting companies must have
a permit to operate in the park.
Near the entry point for the North Fork Canyon Run is
an access road along which visitors can park, as well as a
small parking lot. Visitors come for a popular scenic outlook,
hiking trails and to get on the waiting-list for raft rides when
offered during the spring. The park authority believes that
there is only adequate space for two rafting companies to set
up and operate on any given day.
The first two permit-holding raft companies to arrive in
the morning get exclusive rights to operate for that day.
Different rafters operate on different days, depending upon
their customer bookings and business on larger rivers in the
park.
Daily revenue is determined by the number of rides per
day, which in turn is affected by river conditions. Differences
in water level occur from changes in the volume of snow melt
in the mountains. When the water level is low, the rafts get
stuck or hung-up more often on rocks, reducing the number
of rides. Based on water levels, the two raft operators can
estimate how many trips they’ll make in the day, which is also
affected by their respective decisions to operate one or two
rafts. Four rafts usually operate, two per company, on the
North Fork Canyon run. However, the optimum is usually
three rafts, sometimes two — but hardly ever four rafts in
total.
Let’s take a look at three recurring payoff situations for
rafting businesses sharing the North Fork Canyon Run.
Dominant choices
The outcome for two rafting companies in making a
decision to operate one or two rafts is displayed in the payoff
matrix for the relatively high water level that usually prevails
on the North Fork Canyon run during the spring (Figure 1).
Table 1 shows the number of rides per raft and the effect of
the number of rafts operated on the total payoff for each cell
in the matrix. Although there is a cost in operating an
additional raft, the business owners want to keep their raft
guides or navigators busy. When the first two operators
arrive to establish their claim to operate for the day, each
plans on using two rafts.
The incentive structure of the payoff matrix in Figure 1
creates a dominant choice of two rafts each regardless of what
the other does. Column Rafting Co. has payoffs in the upper
right corner of each cell. The second column has a payoff of
either 20 or 14, which are larger than column one payoffs of
12 and 10 when operating one raft. A column is “dominant”
when the payoff of at least one of its cells is higher and all its
other cells are not lower than the adjacent cells of all other
columns in the matrix. Likewise, for Row Rafting Co., the
payoffs of both cells for operating two rafts dominate the
pay-offs for one raft (lower left corner of each cell).
Combined revenue would be maximized with three rafts in
total. The dominant choices would not have to be made if
the two companies shared the day’s proceeds from operating
three rafts. Such coordination is difficult because there is
usually a different combination of two companies operating
on the North Fork Canyon Run from one day to the next.
When overnight temperatures are lower than normal,
refreezing occurs in the mountains and water levels can drop
enough to reduce the number of raft rides, whether operating
2, 3 or 4 rafts. Figure 2 and Table 2 report the impact of a
lower water level on the number of rides. When more than
two rafts are operated in these conditions, back-ups at the
entry point may cause some customers to leave or other
delays, in contrast to the immediate turnaround when using
two rafts. The incentive structure of pay-offs again produces
dominant choices in the 2nd column and the 2nd row.
However, in this case maximum revenue is the northwest cell,
where each operates one raft. This payoff structure is known
as “the prisoners’ dilemma.”
Volumes have been written about the prisoners’ dilemma
because it focuses on what is lost when participants cannot
communicate. In the original story, two prisoners are in
isolation and both choose to confess, hoping to get a better
outcome. Unfortunately, since they both confess, they each
get the worst payoff. The two prisoners lack the trust in each
other to stick to their “not-guilty” story and mutually get the
better results of the northwest quadrant.
Many studies use the prisoners’ dilemma to discuss the
importance of improving upon worst outcomes through trust
and understanding. But on the North Fork Canyon Run, the
participants in the two-person game differ from day to day,
which can be enough to thwart communication and the
building of trust. As pointed out by Thomas Schelling, it’s
not really a “dilemma” at all but a game of dominated choices
(Thomas Schelling, Strategies of Commitment, 2006, viii).
Contingent choices
Weekends in the park are crowded with sightseers, hikers
and tourists wanting raft rides. The difficulty of finding
convenient parking for the vans and raft trailers increases the
turnaround time when more than two rafts are operated. A
third raft encounters occasional delays but still results in
more rides than if two rafts were operated. A fourth raft
results in a series of delays on crowded weekends. The
decision for each operator about one or two rafts is
contingent upon what the other operator does.
Figure 3 and Table 3 report the payoffs of an incentive
structure that involves contingent choices. Without a
dominant choice, each participant will consider the benefits
of coordinating their decisions to be able to operate three
rafts – one company operates two, the other company
operates one raft, and they split their combined revenue.
The contingent choices of Figure 3 may not always result
in coordinated decisions. The first rafting company to set up
on the North Fork Canyon Run on a weekend day could
choose to operate two rafts on the expectation that the
second company to arrive will prefer to use one raft with a
payoff of 9, as compared to 6 if a fourth raft were added.
This game has a “first-mover” advantage.
The potential success of a first-mover in operating a
second raft depends on the other rafting company’s sensitivity
about fairness. The second raft company could be indignant
about the lack of revenue sharing and decide to operate two
rafts. The first-mover advantage is defeated if raft operators
always react indignantly and chose to operate a second raft.
In that case, the “first-to-arrive” rafting company may choose
to use just one raft so as to secure the pay-off of 12 trips
rather than risk getting only 6.
Clearly, weekends on the North Fork Canyon Run involve
contingent choices that do not come up during the weekdays
when operating two rafts each is the dominant choice for any
two companies.
Challenges in coordinating decisions
Game Theory analysis may appear to be an
oversimplification of actual business decisions, but its purpose
is to highlight cooperation opportunities and their prospects
for success. In each of the three scenarios on the North Fork
Canyon Run there are opportunities to jointly maximize
earnings with coordinated decisions. Although only two
parties have to reach an agreement, the fact that the
companies operating raft trips frequently vary from day to
day diminishes the patience and trust necessary for
negotiating a plan for either three or two rafts.
A payoff matrix reveals the decision cell with the highest
total earnings, but participants estimate their payoffs either
along columns or rows. In other words, each player estimates
its payoffs from a go-it-alone perspective. They don't
compare the total value of each cell or quadrant in a payoff
matrix. This orientation is natural and practical when
thinking in terms of operating alone.
Finally, a Game Theory analysis also demonstrates why
the prospects for coordinating decisions are much improved
if participants are dealing with contingent, rather than
dominant, choices. Participants understand that to choose
for the highest pay-off, they risk getting the lowest pay-off,
while their other choice offers pay-offs between the highest
and lowest. This uncertainty, in contrast to dominant choices,
encourages the parties to seek an agreement on three rafts
with revenue sharing.
The benefits and challenges of coordination in a one-day
encounter may be extended to the longer term of formally
organized cooperatives when the earnings from individual
decisions are interdependent. Market participants are more
likely to become members of a cooperative when their
decisions are contingent upon what other market participants
do than if their choices are dominated.
The lesson from Game Theory is that while businesses
and individuals are committed to their "bottom line," their
goals can be self-defeating if they allow this singular commitment
to create tunnel vision. An eye for opportunities to
cooperate is a useful skill in the pursuit of individual gain.
