Earth Day Game
Department of Mathematics
York College (CUNY)
Jamaica, New York 11451
Suppose two high schools R and C are located near a rectangularly shaped public park which is divided into 4 sectors (numbered 1, 2, 3 and 4) , each a congruent rectangle. The four sectors are laid out as in a 1x4 array. In honor of Earth Day a local business has donated prizes for high school student teams to clear rubbish/debris from the park. The terms of the "contest" are that each school is allowed to have a team of size up to 8 students. Each school gets to assign members of its team to one or more sectors. Since the students are equally diligent in removing rubbish/debris, the "winner" in a sector will be the school that assigned a larger number of students to that sector. When schools assign the same number of students to a sector there is a tie or draw. The payoffs to the teams are found as follows. If a team wins in a sector, that team gets one point (the prize) for winning the sector as well as an additional point because it will have the right to display the school flag in the sector in which it wins for three weeks after Earth Day. When a tie occurs both teams get 0 points. The score for each schools's team sums the results in the individual sectors.
The principals of the two schools act independently. They realize that there are some strategic issues involved with how to assign numbers of students from each school to the sectors and are motivated by school pride to be as successful as possible. Sample actions for school R might be: send 8 students to Sector 3; send 4 students to Sector 2 and 4 students to Sector 4; send 2 students to each sector. What advice would you give the principals?
1. Play this "game" 10 times and record the action you take and your opponent takes in each play. What are your cumulative earnings after 10 plays? What are the cumulative earnings of your opponent after 10 plays?
2. Develop a "notation" for recording what actions you have available and your opponent has available playing this game a single time the payoffs to the two players that result from the various pairs of actions that might be taken.
3. a. Repeat playing this "game" but modified so that the park is laid out as a 2x2 rectangle. Thus, play 10 times and record the action you take and your opponent takes in each play. What are your cumulative earnings after 10 plays? What are the cumulative earnings of your opponent after 10 plays?
b. Does the geometry of the "sectors" affect the way you think about playing as compared with what you did for Question 1 where the geometry of the sectors was different.
4. If trying to understand this game is too "complicated," consider trying the simpler game where the park has only two sectors, a 1x2 layout, and the number of students who can be sent to each sector by each school is 4 students. The payoffs can be arrived at in the same way.
5. Another way of simplifying the game is to assume there is no payoff for being able to display the school flag, only a payoff when one school sends more students to a sector than the other school. Try doing Question 4, 1x2 sectors with teams of 4 with the simplified payoff system.
1. The games described are "symmetric" in that both schools are able to mount teams with equal sizes but one can imagine an asymmetric version where the schools can't mount teams of the same size.
2. Try to find other "settings" or "contexts" where this game might arise from these alternate settings or contexts.