# Predicting Victory in Axis and Allies Revised - Using Logistic Regression

I have created a model that does a good job of predicting who will win in Axis and Allies (Revised edition) - a strategic board game where the outcome depends on a mixture of good tactics, a knowledge of probability, and some luck.

Axis and Allies is perhaps one of the most popular board games in the category of games harder than Risk, but simpler than the hexagon (Avalon Hill-type) war games.

I've been playing it for almost 20 years, and strongly recommend the Revised version as a massive improvement on the old game.

Data Collection
I used data from my own games as well as those from others on the Axis and Allies forum. I collected BattleMap files from after the Russian turn. The BattleMap program provides an easy summary of the unit value of all the land units and naval units of each country. It also says the IPC value of the territory currently occupied.

From those, I created two variables.

IPCDif - The IPC value of Axis territory minus that of the Allies (remember this is measured at the end of the Russian turn)

UnitDif - The IPC value of Axis units minus that of Allied units (also measured at the end of the Russian turn).

As both are measured at the end of the Russian turn, they are generally both negative unless the Axis is far ahead.

I ran a logistic regression using SPSS.

The Model
116 rounds of data - roughly 15 games

Predicted
Allied Win Axis Win Percent Correct
Observed Allied Win 28 10 73.7
Axis Win 5 73 93.6

Overall Percent Correct - 87.1
(It isn't predicting Allied wins as well, because 2/3 of my data was axis wins)

B SE Sig Exp(B)
UnitDif .038 .011 .000 1.039
IPCDif .110 .035 .002 1.116
Constant 6.644 1.501 .000 768.146

Cox and Snell R Squared: .520
Nagelkerke R Squared: .724

I think this means that if UnitDIF changes by 1, your chances of winning change by 3.9%. If IPCDif changes by 1, your chances of winning change by 11.6%.

Also if the IPCDif is zero then it means for the game to be even the UnitDif should be 177 (6.644/.038).

Example
If you are playing the Axis and are behind by 24 IPC in territory, and 8 in IPC unit value.

IPCDif: -24
UnitDif= -8

Odds of winning= e^ (6.644 –24*0.110 –8*0.038)
= e^(3.7)
= 40
(or 40:1)

The 95% confidence intervale for your odds of winning range from 2:1 to 800:1 (roughly e^0.7 to e^6.7).

Analysis
Other factors weren't significant. Bid doesn't matter because its effect is included immediatedly in the UnitDif variable (unless the Axis player takes part of the bid as IPCs, generally 1 IPC to Japan - so a very small amount).

The UnitDif is a stronger predictor than the IPCDif, because the UnitDif measures how much territory the player has occupied in previous turns, as well as how good their strategy is, and how much luck they've received.

UnitDif and IPCDif were highly correlated (bivariate correlation 0.560), thus inflating the standard error for the model (if you exclude IPCDif the standard error for the constant falls by half!).

The model does an excellent job of predicting outcomes. Better than I thought it would! Starting on Round 5, all of the outcomes were correctly predicted. It is only failing to predict who will win early in the game - which is an accurate reflection of the game state, as during the early turns it is hard to tell who is winning.

On the other hand, the sad news is that if you can predict who will win the game by the fifth turn - playing it isn't as fun. I suspect that if I add additional data to my sample that it will include some games where the victor is uncertain in the fifth or later rounds.

### The Formula

To summarize my formula for the probability of the Axis Winning is:

Odds of winning= e^ (6.644 + (IncomeGap *0.110) + (UnitIPCgap * 0.038))

Where e = 2.71828183 (it's a mathematical constant)

^ - is the power of sign, as in x^y.

Two values that must be measured at the end of the Russian turn (just after Russian unit placement):
TerritoryGap = Axis Territory (in IPCs) - Allies Territory

UnitIPCGap= total value of Axis units (in IPCs) - total value of Allies units (includes AA guns, doesn't include ICs).

The result is an odds ratio. So if you get 19 - it means the Axis has a 19:1 odds of winning (95%). If you get 0.1 it means the odds are 1:10, or their chance of winning is 1/11).

--- An Example ---
Commonly at end of the first Russian turn (R1), you will have Russia taking WRU and BEL. In this case:

TerritoryGap = (36+30) - (28 + 30 + 42) = -34
A typical IPC gap = -75

Axis Odds = e^ (6.644 –34*0.110 –75*0.038)
= e^ (6.644 -3.74 - 2.85)
= e ^ (0.054)
= 1.055
The Axis Odds of winning are 1:1.055 or 1/2.055 or 48.7%

Which is what you'd expect after the first round of play.

-- Confidence Interval --

One problem with the result is that there is still a relatively big 95% confidence interval. So your actual odds could be as much as roughly 7 times more, or 7 times less those that the model predicts (the standard deviation for the constant is 1.5, two standard deviations are 3.0 and e^2 is roughly 7).