THE HURRICANE GAME

Unpublished article by Jerry Tuttle, FCAS, CPCU

Platinum Underwriters Reinsurance Company

The NCTM’s Standards for K-4 encourage active involvement in doing mathematics, the development of mathematical thinking and reasoning abilities, the application of mathematics to real-world problems, and the inclusion of a broad range of content (1989 pp. 17-19). I have accomplished all of this in a mathematical simulation called The Hurricane Game that I would like to share with you. I have tested this game with classes of first and third graders. I specifically chose primary grade students to play this game, feeling that they are ready to be exposed to this kind of mathematical thinking.

I divide the class into teams of two students. I pretend that the students are now adults, and as such they earn a salary and own a house. I pay them an annual salary of $100 in money from the game Monopoly, and I give them a Monopoly house. (Of course, students know $100 is not a realistic salary for an adult, but I want to keep the numbers fairly simple.) I then give them the bad news: sometimes there are hurricanes, and hurricanes may cause great damage to people’s houses. The main issue in the game is whether or not the students should buy hurricane insurance.

At this point I pause and ask the students if they know the names of any hurricanes. The major US hurricane in recent years was Hurricane Katrina in 2005, which some students know from social studies and television. The entire insurance industry paid over $41 billion (Insurance Information Institute) for this catastrophe; other damage was not insured. I write down the number of zeroes in a million and a billion.

For the first round of play I tell the students that major hurricanes occur about once every three years and that when one occurs it may cause no damage, small damage, or total damage. In this first round I offer to sell hurricane insurance for $50. The students have to decide as a team whether to buy it or not. This is not unlike a real-world decision that adults have to make. There is the additional real-world consideration that many decisions are made by groups and not every member of the group agrees with the decision.

(I am making a number of simplifications with the real-world situation. A standard homeowner’s insurance policy provides protection against hurricanes. Homeowner’s insurance is not required by law, but it is commonly required by banks as a co-requisite to obtaining a mortgage. So in practice an adult does not really decide whether or not to buy hurricane insurance. A better insurance example might be whether or not to buy automobile theft and collision insurance after the car loan is paid off, but I chose not to use that example because automobile claims generally do not affect many individuals simultaneously, except when there are disasters like hurricanes. Also, I ignore the reality that insurers charge different premiums to different individuals based on various presumed claims-producing characteristics.)

I spend some time emphasizing that there is neither a right nor a wrong decision. Although older students who have learned expected values may be able to make an expected value decision (the expected value with insurance versus without insurance), in the real world we do not always make expected value decisions. In particular, your wealth level, your tendency to like or dislike taking risk, and the variability of the possible outcomes may influence your decision. (Bowers 1986, p. 3)

Another possibility for older students is to have some of the students assume the role of the insurance company, where they will have to decide how much premium to charge.

The students get several minutes to decide whether or not to buy the insurance. I ask them to explain their decision. A few students make reference to the "once every three years," implying that the hurricane probably either will or will not occur this year. Our group is about half risk-takers who don’t buy insurance and half risk-avoiders who buy the insurance. When students buy the insurance, I ask them how much change they get from a hundred dollar bill. This takes some thought; many children who can do 10-5=5 have some difficulty with 100-50=50. I give them change, and I give them a Monopoly deed as proof of insurance.

One of the students then rolls one die to determine whether there will be a hurricane. I take a moment to examine a die, and I ask the class what numbers can occur and whether any number is more likely than another. The die roll was four. I announce that there was no hurricane that year.

In response to a student’s question, I discuss why the insurance company does not refund your premium in a year when you have no claim. The class understands that the insurance company is in effect saving its money to pay for the year when a hurricane occurs (literally saving for a rainy day). Some insurance companies are unsuccessful in doing that; eight of them became insolvent when they could not pay their Hurricane Andrew claims (Snyder 1993, p. 107).

In year two I give the teams another $100 in salary and I sell them hurricane insurance for forty dollars. Again I ask the students how much change they will get. A few teams who bought insurance in year one do not buy it in year two. Some students say that they feel the chance of a hurricane in year two has decreased because there was no hurricane in year one. A subsequent lesson could discuss independent versus conditional probability.

A student rolls a die to determine whether there will be a hurricane, and the roll was two. I announce that this means a hurricane. I then explain how we will determine each team’s amount of damage. Each team will roll two dice. If the sum is 2, 3, 4, or 5 there is no damage. If the sum is 6 or 7 there is $40 of damage. If the sum is 8, 9 ,10, 11, or 12 there is total damage to the house, which I define as $300. (Students did not seem to object to the fact that they were not told of these probabilities when they made their insurance decision; however, in the real world it is rare to be given the probability estimates before making a decision.) A subsequent lesson could discuss how many different sums two dice can make and why the probabilities are different.

Each team rolls the dice and adds the sum. Some students count the dots by ones, and others add the two numbers mentally. Teams who bought the insurance will be paid by the insurance company for any damage, and teams who did not must pay for the damage out of their own funds. There is more money to exchange and change to make. A few uninsured teams had total damage, and the insurance company graciously but unrealistically offered to loan them money to pay for the damage so they could remain in the game, but the loan must be repaid.

This is a perfect opportunity to discuss negative numbers! I discuss why a team that owes $150 that it doesn’t have is worse off than a team that might have zero dollars, although they look the same.

In year 3 I sell insurance for $60. This time I explain a little more. A die roll of either two or six will result in a hurricane. This is equivalent to once every three years. Each team is paid its salary, but teams that owe money must repay their debt before buying insurance. A team that still owes money must sit out the round. Excluding the teams that sit out, nearly every team buys insurance this time. One team that doesn’t explains its decision by saying that there couldn’t be two hurricanes in a row. Mathematically, that team is wrong; a student certainly could roll a two twice in a row. Meteorologically, I suspect that team is wrong as well, but the science of hurricanes is beyond the scope of this lesson.

The mathematical model that I am using is a simplified version of the collective risk model (Bowers 1986, p. 317), where the number N of claims occurring in a year is a random variable, and where there are N individual claim amounts X₁, X₂, …, X_N which are also random variables. The individual is concerned with the sum of the X’s, which is the total amount of claims in a year. In our game I set N=1; I did not consider the real-world possibility of more than one hurricane in a year.

Year 3 did not have a hurricane. In year 4 I reduce the premium to $50 and announce that this will be the last round. The previously bankrupt teams are back in the game. One team is down to its last $50. They struggle with the question of whether to spend their last $50 on insurance and end the game with zero, or whether to go uninsured and risk a potential loss of either $50 or $300. Another team who bought insurance in year 4 was audibly rooting for a no hurricane roll. I ask whether they really care; having bought insurance, the amount of money they will have left is the same whether there is a hurricane or not. Peace of mind is one of the reasons people really do buy insurance.

After round 4 the teams add their money and we see who "won." The team that won did not buy insurance in several years, including the year of the hurricane, but did not have any damage that year. In the short-run, if you are lucky, not buying insurance is of course the optimal strategy. This is true in the real world as well. Assuming you and your insurance company agree on the amount of your expected claims, then not buying insurance will save you the amount of the premium representing the insurer’s expense and profit loading.

I invite each student to tell me what he or she learned. Most students talked about hurricanes or insurance. A few talked about spending large amounts of money and subtracting from a hundred dollar bill. The students also wrote about the simulation in their math journals later that day.

In summary, we have simulated a real-world problem and gotten primary grade students to make some non-rote decisions and to do some mathematics including making change, estimating probabilities, and thinking about such things as independent probabilities and negative numbers.

Bowers, Newton L. et al. Actuarial Mathematics. Itasca, IL.: The Society of Actuaries, 1986.

Insurance Information Institute. www.iii.org/media/facts/statsbyissue/catastrophes

National Council on Teachers of Mathematics, Commission on Standards for School Mathematics. Curriculum and Evaluation Standards for School Mathematics. Reston, VA: The Council, 1989.