# Dice Games - Redefining Winning

Who wins a game? It depends totally on how winning is defined. What will happen if players use inconsistent definitions of winning? This simple experiment will demonstrate ideas that may be generalized to understanding voting, sports and economics.

Objective: Students will use mathematics to question the definition of winning. Students will analyze experimental results using methods of probability.

Readings: Beyond Numeracy - voting systems chapter, Mathematician Reads the Newspaper. chapt.1 voting, power, and mathematics

Reformatted: January 2010

Many simple games exist whose outcome is dependent on chance. Here's two arrangements where both players may win, or both may lose.

Materials: 2 blank dice of different colors - to be renumbered

Method:

• We will call the dice red and white.
• Number the Red die with 1,2,3,3,6,6.
• Number the white die with 1,1,1,2,8,12.
• Have two players select a die and compete for the best roll.
• The first player should keep score by tallying the winner of each roll.
• The second player should keep score by tallying the total points that each player scores in each roll.
• They should play at least 20 rounds.
• At the end, they should compare who won by each method.

Sample Rounds:

•  Round Rolls Scoring First Method Scoring Second Method Red Die White Die Red Player White Player Red Player White Player 1 3 1 1 0 3 1 2 2 2 1 0 5 3 3 6 8 1 1 11 11 4 3 1 2 1 14 12 5 6 12 2 2 20 24 In our example, by the end of the fifth round the player using the first method thinks they have tied 2-2, the player using the second method thinks white has won 24 point to 20.

Related discussions at this site

Analysis:

One of the preferred methods of analyzing probability is to make a chart. Both games will be analyzed using two 6x6 chart showing all of the possible roll combinations. And the possible scoring combinations.

Scoring Method 1: show who gets the point for the round
 Rd \ Wh 1 1 1 2 8 12 1 tie tie tie White White White 2 Red Red Red tie White White 3 Red Red Red Red White White 3 Red Red Red Red White White 6 Red Red Red Red White White 6 Red Red Red Red White White
From this chart, we can see that in a typical set of 36 rounds red will win about 19 times, and white will win about 12 time, with 6 ties.

Scoring Method 2: show the difference in scores for each round
 Rd \ Wh 1 1 1 2 8 12 1 0 0 0 1 7 11 2 1 1 1 0 6 10 3 2 2 2 1 5 9 3 2 2 2 1 5 9 6 5 5 5 4 2 6 6 5 5 5 4 2 6
If we add up all the differences in this chart, we can see that in a typical set of 36 rounds white will beat red by 24 points.

Extensions:

Now that students have seen the same game have different outcomes depending on how you score it, they need to generalize this concept to common life experiences.

• We may have noticed that the die with the higher median wins more rounds; the die with the higher average wins more points. What does this tell us about interpreting statistical data?
• Is it possible in basketball, football, volleyball, etc. for the team with the most points not to make the playoffs?
• Is it possible in the American two party system for the candidate with the most popular votes not to become president? Has this happened? If so, how often?
• If there were three presidential candidates in America, under what circumstances could the candidate with the least popular votes become president?
• Using the books referenced at the top of this page students can research how this applies to democratic elections.
• Since top 10 and top 40 countdowns are popular with kids they should be able to show how different scoring methods give totally different results. For example a count down that asks people to name their favorite song will give different results than one that asks people to name their favorite three songs giving 10 points to the first 5 to the second and 3 to the third, which will surely give different results than asking people to list their favorite song and their most hated song.
• For a research project, how did the computer program called "tit-for-tat" win an entire competition without ever winning a single round? What does this imply about cooperation in human relationships?