My wife’s family is huge. She is one of eight, and many of her siblings have carried on this tradition in their families as well, the result being that her parents have over 30 grandkids.
This provides me ample opportunity to play all sorts of games. On our recent annual vacation to a camping cabin “resort” in the north woods of Minnesota (where cell phone reception is spotty at best), on one of those days when we are trapped indoors because the rain prevents any playground, beach, biking, fishing, or golf activities, I settled down to a game of Monopoly with an 8-year old.
The problem is…I should have been thinking more than I was.
Monopoly is a board game that takes you around a square with 10 spaces on each side, for a total of 40 spaces in all. Most of the spaces represent “properties”, which you can buy and own, and when other players land on them you collect “rent” from them. Each turn you roll 2 dice and move your token accordingly. If you land on someone else’s property, you must pay them the rent for that property.
Most of the properties belong to color groups of 2 or 3. If you own all the properties in a color group, you have a “road”. When this occurs, you are able to invest in houses and hotels, which significantly increases the rental income you collect from other players. The only exception to this color group is the railroad and utility groups, which cannot be improved upon, though by owning more than one rental income increases.
It is only with a large amount of luck that you can obtain a road on your own movements. Because of this, at some point in the game players start to make trades – combinations of one or more properties and perhaps cash in exchange for others.
After having traversed the board quite a few times, we arrived at the situation where all the properties were purchased but nobody owned a road. The 8-year old I was playing with proposed a trade, whereby I would give him the two railroads I owned (he owned the other 2) in exchange for Pacific Avenue, one of the three properties making up the “green” color group. Since I owned Pennsylvania Avenue and North Carolina Avenue (the other two greens), I would be in a position to build houses and hotels to increase my income while others would not.
Given the properties I held, it was also not possible for anyone else to get a road, so this trade appeared to me to be one where I would be able to slowly establish a juggernaut that all would succumb to.
Unfortunately for me, I did not take “Expected Value” into account, whereby my 8-year old opponent did (though maybe not consciously).
What is Expected Value?
For example, if I receive ²1 ( the symbol ² stands for Treasury Cafe Monetary Units, or TCMU's, freely exchangeable into any currency of your choosing at any exchange rate you desire) should a coin flip result in heads, and pay ²1 should the coin flip result in tails, given a 50/50 chance of each occurring, then my "Expected Value" is ²0 (.5 * 1 + .5 * (-1) = 0).
What if I receive ²1 on heads and pay ²0.50 on tails? Then my expected value is ²0.25 (.5 * 1 + .5 * (-.5) = 0.25).
What if I receive ²1 on heads and pay ²0 on tails? Then my expected value is ²0.50 (.5 * 1 + .5 * 0 = .5).
Applying Expected Value
Now that we understand expected value, we can apply this knowledge to my Monopoly trade with my 8-year old opponent.
The Monopoly board has 40 spaces, so if we assume that landing on each one is equally likely, then the probability of landing on each space is simply 1/40, or 0.025. After the trade, my opponent will have 4 railroad properties, each requiring a payment of 200 from the person landing on them. Using Formula B, they will have an expected value of 20 (.025 * 200 + .025 * 200 + .025 * 200 + .025 * 200 = 20).
I had the funds to put up one house on my green properties. Those landing on green properties with one house need to pay rent of 130 for two of the three and 150 for the other. Thus, my expected value, using the formula in Figure B, is 11.5 (.025 * 130 + .025 * 130 + .025 * 150 = 10.25).
Since we were playing each other, my opponents expected value is also my expected payment, and vice versa. Unfortunately for me, this means that I can expect to pay 20 while receiving only 10.25, and thus my net expected value is -9.75. Because of this, the possibility of amassing enough cash to buy another round of houses for my green properties (which would require 450) is quite unlikely. If I could achieve this, it would put me in a positive position, as the expected value with two houses is 30.75 (0.25 * 390 + 0.25 * 390 + 0.25 * 450 = 30.75).
The lesson from this basic analysis was that in order for me to make the trade, I needed enough cash on hand to build two rounds of houses on the green properties immediately in order to make a positive expected value. Lacking that, I should not have made the trade.
Since movement in Monopoly is governed by the roll of 2 die, the 1/40 probability assumption we used in the last section is somewhat inaccurate. If our token is on the Board in space #1, then it is more likely that on the next roll we will land on space #8 (i.e. rolling a 7) rather than space #3 (i.e rolling a 2), so each of these spaces have different probabilities (the odds of a 7 are 7/36, while those of a 2 are 1/36).
Similarly, on the next turn, the probabilities of different properties being landed on will depend on where we landed the turn before. Had we rolled a 2 last time and moved to space #3, on our next turn it is now more likely we will land on space #10 (rolling a 7) instead of space #15 (rolling a 12).
This concept, that the future outcome is determined by the past, is known by the term “path dependence”.
A Trip to Monte-Carlo
One solution to estimating results in a path dependent situation is to perform a Monte-Carlo simulation. Monte-Carlo models use statistically based random numbers to project the future over and over. By doing this, we can develop an estimate of the probabilities of events we are concerned about occurring.
In order to accomplish the simulation, we program the movement process around the Monopoly board, taking into account the die roll (in this setting the random element of the Monte Carlo), and the game elements that impact position (e.g. the “Go to Jail” space, Chance and Community Chest cards, etc.). This was done using a combination of Excel and Visual Basic for Applications (I am happy to email this spreadsheet and code to you if you’d like, simply connect with me on LinkedIn and provide me an email address).
We then simulate 1,000 games from every one of the 40 possible starting positions. In each simulation, all players had to go around the board at least 5 times. This results in 40,000 data elements to analyze.
Evaluating the Data
The t-test is a statistical metric that determines whether the average of one set of data is significantly different (meaning likelihood is set to a high threshold, such as less than 5% chance) than the average of another set. Figure C shows the formula for a t-test statistic for equal sample sizes with an assumed equal variance.
One thing I like to do as an analyst is verify that my understanding of equations is sound and that the programs I am using are performing calculations according to that understanding. For that reason, I calculated in Excel the t statistic for the test between Expected Value results for starting position #1 and starting position #16 (Figure D), and then compared that to the R output (Figure E).
Looking at the orange line (starting position is “Go”, space #1), there is not a significant difference in Expected Values for this position vs. its near neighbors (up to around space #10, and space #30 and up) but is significantly different from Expected Values in the 10’s and 20’s.
Conversely, the brown line (starting position “Pennsylvania Railroad”, space #16) shows no significant difference between its near neighbors but significant difference from starting spaces further away (spaces #1-#10 and #30 +).
This process confirms the path dependency of Expected Value, they are different depending on where you start are on the board.
The mean of the Expected Values in Figure G is 10.18, surprisingly close to our initial pass estimate of 10.25. So while path dependence does occur in the game, in this case it is not great enough to affect the outcome over a simpler set of assumptions.
Finally, I looked at the number of times the simulation resulted in a higher expected value for my side of the trade vs. that of my 8-year old opponent. On average, across all starting positions, only 13% of the time did I come out ahead (ranging between 10% and 16% depending on starting position). Based on this, I was very unlikely to win the game.
My failure to utilize the tool of Expected Value cost me the game. Sitting in a cabin near a lake in the woods is not the first place one might think to utilize analytical tools, however, in this case it would have helped.
The tools we have learned and deploy can often be used in more settings than we might think, so long as we are willing to be a little creative with them.
Next time I play that kid I might bring my computer!
Calculating Expected Value is a tool that can be used to assess alternative situations in order to inform decisions about what to do or not. Through the use of Monte Carlo simulation, even situations involving path dependent factors can utilize the Expected Value tool. As always, judgment needs to be utilized in assessing data and output of these calculations.
· Have you encountered game situations where statistical concepts have been useful?
· How have you deployed analytic tools in unusual situations or settings?
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