Each team has 100 points to distribute over defense (DA, or DB), midfield (MA or MB), and attack (AA or AB). After that the game is played. It consists of 9 (changeable) identical two-step rounds. In the first step it is decided into which half the ball goes. This is decided randomly with odds being MA : MB. That means the probability that A will attack is MA/(MA+MB), and the probability for B attacking is MB/(MA+MB). In the second step of each round it is decided (again randomly) whether there will be a goal for the attacking team. The odds for a goal are AA : DB (probability for a goal is AA/(AA+DB)) if A is attacking, and AB : DA if B is attacking.
Your team is team A. Distribute the numbers in the fields. Press the
button to get a fair sum of 100.
Now you can either press 18 times the
button
to see all steps of all 9 rounds, or finish the whole game
with the 
button.
You can also simulate 10, 100 or 1000 games (taks some time).
The results are summarized in the (yellow) table.
What outcome would we expect?
See here for an analysis of the simulation.
For our purposes it suffices to state that, given two teams
a = (DA,MA,AA)
and b = (DB,MB,AB),
it s possible to compute the probabilities
pw(a,b) for a win for team A,
pd(a,b) for a draw, and
pl(a,b) for A losing the game.
Since exactly one of these alternatives must occur,
pw(a,b)+pd(a,b)+pl(a,b)=100%.
Furthermore, the fraction of possible points that team A is expected to get is
ep(a,b) = 3*pw(a,b)+pd(a,b).
All these expected values are computed for the given teams in the light blue part of the table above
if you click the 
button.
Exercise 1: Choose a distribution for your team A. Compute the expected values. Now play 10 games against team B. Compare actual and expected values. Are they close? Now play 100 more games and compare. What about if you play 1000 more games? Express your observation concerning closeness versus number of simulations. Solution
Exercise 2: Choose distribution (40-40-20) for your team A and (30-40-30) for your opponent B. The expected values of points for team A is 44.89%. Does that mean that team B is expected to win 55.11% of the possible points against team A? Check your answer by interchanging team A and B. Explain. Solution
We assume that we may choose our distribution after we know the distribution
of our opponent B. Of course, this more information should allow us to find
a very useful distribution. We can, in principle, compute these expected values
for each distribution. Then we choose that distribution for our team A which
maximizes the expected number of points. Press this button 
to improve the expected value of points of your team. Doing it several times
you get a distribution hopefully close to the optimum.
Each team X has three coordinates, defense number DX, midfield number MX, attack number AX. But since the numbers must add up to 100, two of them, say DX and AX are enough. Therefore we can visualize the teams as points in the plane.
Exercise 3: Choose 5 different B teams B1, B2, B3, B4, B5. Visualze them on the plane. For each team, find a more-or-less optimal A opponent, and draw also these A-teams A1, A2, A3, A4, A5 in the plane. Can you formulate any conclusion on how to choose a team when facing a given B-team? Solution
Open question: For each point y = (DB,AB), we define the strategy a(y) to be the point (DA,AA) where the strategy (DA,100-DA-AA,AA) achieves the best possible expected value against opponent (DB,100-DB-AB,AB) with respect to the number of points. This is a mapping from the triangle {(x1,x2)/0 < x1 < 100, 0 < x2 < 100, x1 + x2 < 100)} into itself. Is this mapping continuous? Is it 1-1? Is it surjective?
The team president wants some guarantee like: "Against ANY team we expect to get
at least 30% of the (3) possible points". To obtain these numbers, and to find the
team that hurts your team A the most, use the
button (several times, or the "10*bad"
option).
Exercise 4: Is that team that can hurt your team the most also that optimum team getting the highest expected point number against your team? Look at examples. What has the modus of how many points you get for a win (which has been changed in the eighties from 2 to 3---you can change this value as well in the table above) to do with your answer? Solution
How would you maximize the guarantee score? It is the same problem as if you play against your neighbor, but now he insists that both of you have to reveal their distribution simultanuously. How do you choose your distribution?
Exercise 5: Choose 5 different possibilities A1, A2, A3, A4, A5 for your team. Visualze them on the plane. For each, give the guarantee. Can you draw any conclusion on how to build your team A if a high guarantee value is desired (by the president)? What if the midfield number MA is fixed for some reason? Solution
We have seen how to design your team A when facing one fixed team B. Your team may be more active than passive. We have also seen how to design your team A when you want to maximize the expected value of points over all possible teams. How would you design your team A when facing several, but fixed and known other teams? Go to the Champions League? for this question.
Here (and on all other pages) we use The Central Randomizer 1.3 (C) 1997 by Paul Houle (houle@msc.cornell.edu).