U.S. Entry

[paulderynck]

Back in early April or thereabouts, Steve posted his code here for the selection of U.S. Entry chits. He proposed using the same chits as in the boardgame, but wanted to keep the pool of possible numbers infinite rather than finite. Meaning in WIF for example; once you draw all the zeros you can't draw any more, but as proposed for MWIF - you could. This was discussed a little in this forum and more intensely in the beta test forum. Essentially, some of us were concerned that the US Entry system might be skewed by making the chit draw pool infinite.

After discussion, Steve approved Ullern and I doing a study into the performance of the possible chit distributions, and if warranted, providing a recommendation for a different infinite distribution to use in MWIF which would more closely follow the performance of the finite pool of chits in WIF. Both Ullern and I spent many hours on this project and finally provided a report to Steve on July 3. Steve suggested posting the report, so here it is:

(The formatting may be a bit odd as the forum does not handle Word tables very well so some of this is jpegs.)

Report of Study into Method of Generating US Entry Chits for MWIF

Abstract

The assumption is that an infinite pool of US Entry chits leads to disparity between accumulated US Entry totals achieved for MWIF as proposed, versus the board game of WiF. Using three different methodologies, the assumption is confirmed to be correct, and using the same methodologies a new set of US Entry Chits are developed to more closely match the MWIF infinite pool to the results of the board game WIF. The recommended distribution is:

[paulderynck]

continued...

Authors:

Paul and Ullern.
Monte Carlo simulation program by Paul.
Statistical approach by Ullern.
Statistical analysis of Monte Carlo target results by Ullern.
Benchmarking by Paul.
Campaigns scripts mostly by Ullern some by Paul and a lot of input from people at the MWIF Beta tester list.
Also credits to Steve for help on lots of different issues along the way and for defining the general resolution process.


Methodology:

The chit draws in a game of WiF were first simulated using a Monte Carlo simulation. The turn by turn draws were based on a scripting file for a particular strategy. 26 different strategies were scripted to be used this way. An initial analysis of the results proved that any suggested discrete distribution would have to accommodate three basic strategic choices: Barbarossa in 1941, Barbarossa in 1942, or Sitzkrieg. This is because in trying to achieve a similar result as the finite distribution in WiF, these three choices impact the US Entry total differently because of the chits held in the pact pool(s). It became an exercise in trying to make “one size fits all”.

For producing the final results, three different methodologies were used: The Monte Carlo simulation, the Statistical approach and the Benchmarking approach.

The Monte Carlo simulation draws random chits and when premade targets are reached, like the ability for the U.S. to make a Declaration of War (DoW), the WIF turn this happens is recorded. Doing this 100,000 times, the results are sample distributions of when the targets are reached.

The Statistical approach is to use statistics formulas and data to calculate the expected value for a pool of chits drawn from the common pool. Using the 26 scripted campaigns, the number of chits needed for the U.S. to DoW is calculated, by adding chits one by one until the mean of the distribution hits the necessary target number. The statistical approach was used to estimate one target only, a U.S. DoW (likely to be followed soon thereafter by a second DoW on the remaining part of the Axis). This is labeled “Full War”.

(An observant reader would note that there is a difference between the Statistical approach and the Monte Carlo simulation since a result from the Monte Carlo simulation is a distribution of turns for when a target is reached, while a result from the Statistical approach is a distribution of chit values at the turn where the mean of the distribution exceeds the target value and then this turn is recorded for comparison.)

The Benchmarking approach is also using the Monte Carlo simulation program, but using a “benchmark” set of scenarios instead of scripted strategies. The objective was to observe the effects of the finite (WIF) versus infinite (MWIF as proposed) distributions as well as the effect of pacts, while holding the number of chits drawn constant. Two benchmarks were established (high and low number of chit draws; high chit draws are 3 to start then 2 per turn until JF42, then 4 per turn from then on; while low chit draws are 3 to start then 1 per turn until JF42, then 2 per turn from then on) for the three basic strategies (Barb 41, Barb 42 and Sitz) combined with three pact possibilities (no Russia-Japan pact, early Russia-Japan pact and midway Russia-Japan pact). This yielded 18 Benchmark runs.

In order to recommend an alternate infinite distribution, it was necessary to weight the results for the three basic strategic choices. After discussion of the preliminary results; the choice was to weight all Barb 41 results 50%, all Barb 42 results 45% and all Sitz results 5%.

To arrive at a recommendation was an iterative process. The Statistical approach also showed what the expected mean of the common US-entry pool is at different times during a board game of WIF. This knowledge was used to make a first guess for what the new recommended MWIF distribution should look like. From there on, small changes to the distribution were made, in a trial and error process, to improve the results from the Monte Carlo simulation.


Results:

The results of the Benchmarking approach are:

1. In all cases MWIF as proposed was giving a lower accumulated US Entry total than WIF was, for any given scenario.

2. In the boardgame with finite draws, it is true that a later Barb and/or a Sitz results in a higher accumulated US Entry total over the same number of turns then a Barb 41 if chit draws are held constant. However, it should be recognized that in a real game of WIF, players have two choices when indulging in a Barb 42 or a Sitz strategy. They are either aggressive or passive. If aggressive, then the resulting chit draws are bound to drive up the US Entry total (as illustrated in the actual WiF scenarios we ran for these strategies). If passive, then obviously fewer chits are generated then even in the benchmark runs and US Entry will be lower compared to a Barb 41.

The final iteration was made using the Monte Carlo simulation and the following recommendation (called the ‘PN4’ distribution) yielded the best match to the annual averages of the WiF chits based on the 50-45-5 weights of the scenarios.

[paulderynck]

continued...



The Statistical approach verifies the Monte Carlo results. The Statistical approach gives results that are in between 0.7 turns early and 0.2 turns late compared to the Monte Carlo results. (If fractional chits are drawn to reach the target exactly, the Statistical approach gives results that are in between 0.8 turns early and zero turns late.) This difference is likely due to the fact that the Statistical approach does not truly measure the same distribution as the Monte Carlo simulation (see methodology).


Conclusion:

It is evident from these results that the recommended distribution is a much superior match to the performance of the WiF chits than the original infinite distribution was. The final target is the hardest to match but generally is within half a turn of what WiF would produce. That half turn is early for Barb 41s and late for Barb 42s and Sitzs.

[paulderynck]

Here is an example of one of the charts from the Monte Carlo simulation. The Axis strategy simulated was to Close the Med and then do a Sitzkrieg (meaning try to hold the Nazi Soviet Pact as long as possible and build a formidable defence to outlast the allies through to the end of the game). The chart shows the spread of turns the U.S. can choose the War Appropriations option for 100,000 trials. The WiF (blue) is the original boardgame with a finite pool of chits. The "Inf" (purple) is those chits in an infinite pool as proposed originally and the "PN4" (cream) is the recommended distribution for MWIF.

Note the better correlation of the PN4 distribution to the original WIF performance.

With 26 strategies and 5 targets plus the benchmark runs, there are 220 charts like this in the study.

[paulderynck]

Here is one more example. The Axis strategy is the "Bessarabia Gambit" meaning to DoW Yugoslavia on Impulse 3 of SO39 and align Rumania so that Russia cannot demand Bessarabia. Its a bit of a rules loophole but some WIF players do it. Shown here is when the U.S. can expect to Gear-up. When doing this, the Axis tries to do a Barbarosa in 1941 and that was assumed for the chit draws.

[paulderynck]

The picture below shows how the proposed infinite pool would have deviated from the finite pool of WIF chits. This is from the Benchmark runs and is the most extreme case. The lower lines are the low benchmark run and the higher two are the high benchmark run. The chart shows the average per turn accummulated U.S. Entry total for a generic strategy where there is also a Russia-Japan non-aggression pact from SO39 onward and the Axis is doing a Sitzkrieg. In other words, there would be lots of chits in the two pact pools.

As stated, this case has the greatest divergence between the WiF boardgame chit totals and the proposed infinite drawing of those chits, but the same effect was observed in all the benchmark runs.

[MajorDude]

Wow!

Superb work here!

Bottom line though - good to keep them finite or go infinite?

Usually, if you envelop your variables into a finite set, by this very act you make impossible an infinite set without altering your 'finite' set of 'possible' outcomes...

[paulderynck]

ORIGINAL: MajorDude

Wow!

Superb work here!

Bottom line though - good to keep them finite or go infinite?

Usually, if you envelop your variables into a finite set, by this very act you make impossible an infinite set without altering your 'finite' set of 'possible' outcomes...

Steve wanted the draw to be infinite (i.e. an unlimited pool) so that decision was already made. What we did was to provide a distribution that could be drawn from an unlimited pool and would perform much closer to the way the finite pool in the WiF boardgame performs.

The discussion about programming MWIF to work with a finite pool just like WIF does had already happened on the beta test side. That method was distasteful to Steve and his was the designer's decision, so no point in further debate there.

Essentially the quantities of chits in the PN4 table translate to the odds of drawing a chit of that value thoughout that year whenever chits are drawn. With the finite pool, those odds change continuously by a factor related to the chit you just drew.

[Ullern]

ORIGINAL: MajorDude

Wow!

Superb work here!

Bottom line though - good to keep them finite or go infinite?

Usually, if you envelop your variables into a finite set, by this very act you make impossible an infinite set without altering your 'finite' set of 'possible' outcomes...

Thanks.
The MWIF distribution will be infinite. As you can read from the conclusion above the PN4 distribution which we suggest have an average for entering the war within half a turn compared to the original board game. But can be more in some cases, and a 1941 Barbarossa have an average of 0.7 turn error. But the Standard deviation for the original game was in the ranges 0.39 turns to 0.8 turns, so this is all within standard deviation and as good as we can hope for when we are going to match an infinite and a finite distribution.

[Froonp]

ORIGINAL: MajorDude
Superb work here!

Bottom line though - good to keep them finite or go infinite?

Usually, if you envelop your variables into a finite set, by this very act you make impossible an infinite set without altering your 'finite' set of 'possible' outcomes...

Good question finaly.

1) The finite distribution don't please Steve, but is a perfect match to WiF FE, and is also a perfect match to any variation WiF FE may have in the future, or to any variation that one group of player would like to try on the game.

2) The infinite distribution pleases Steve, but is not a perfect match to WiF FE, which is very bad as US Entry is pivotal to the play balance of the game. To make it a perfect match to WiF FE, Paul and Ullern had to do 1,5 months of calculations and statistical modeling, and thanks for their efforts, will will have in MWiF the same US Entry behavior as in WiF FE, so the same play balance in this regard. But any future adjustement will be a nightmare to make, as casual people are incapable of doing the analysis that Paul & Ullern have done, so we will be loosing control on any future adjustement possibility.


Let me develop.
The US Entry in WiF FE is such a delicate balance that it was tweaked twice in the WiF FE game's history. In 1996 was published the Final Edition of the game, with Countersheet (CD) 24 and 14 providing the US Entry markers.

In 2000, a new CS24 was published, it was the first tweaking. It was reported that US Entry was too variable. On this new CS24, 5 US Entry markers (3 of 1939 and 2 of 1940) had their value modified (some decreased some increased) :
1939
0 --> 1
3 --> 2
5 --> 4

1940
0 --> 1
0 --> 1
These modifications had the effect of tightening the distribution (std decreased) of chits around the average that decreased for 1939 and increased for 1940, thus giving a net result of less variability.


In 2007, a new CS24 was published, this was the second tweaking. On this new CS24, 7 US Entry markers (2 of 1939, 4 of 1940 and 1 of 1941) had their value modified (some decreased some increased) :
1939
0 --> 1
0 --> 1

1940
0 --> 1
0 --> 1
3 --> 2
4 --> 3

1941
3 --> 4
These modifications had the effect of again tightening the distribution of chits (std decreased) around the average that increased for 1939 did not change for 1940 and increased for 1941.


Manipulation of US Entry is manipulation to the play balance of the game.
Thanks to Paul and Ullern, we will have the chance to have MWiF behave very similary to WiF FE here.


But what will be done to MWiF if in the future it is discovered by the WiF FE developpment team that the US Entry needs another tweaking ?
US Entry tweaking is one of the central discussions amongst WiF players around the world (and it is so amongst people involved in the design of WiF FE), and embedding in the design of MWiF a factor that prevent any future modification only by specialists, is not desirable IMO.

I'm not saying that US Entry should be moddable at will by casual players, I'm just saying that we should keep in the design of MWiF a way of having it modified in the future so that it keeps matching WiF FE, to take advantage of any progress made in this section of the game.

[BallyJ]

I am so glad this issue has been revisited. It has been one of my major concerns.Froonp you raise a valid point IMO.What ever the decision my thanks to those who did the investigation.

[Shannon V. OKeets]

ORIGINAL: Froonp

ORIGINAL: MajorDude
Superb work here!

Bottom line though - good to keep them finite or go infinite?

Usually, if you envelop your variables into a finite set, by this very act you make impossible an infinite set without altering your 'finite' set of 'possible' outcomes...

Good question finaly.

1) The finite distribution don't please Steve, but is a perfect match to WiF FE, and is also a perfect match to any variation WiF FE may have in the future, or to any variation that one group of player would like to try on the game.

2) The infinite distribution pleases Steve, but is not a perfect match to WiF FE, which is very bad as US Entry is pivotal to the play balance of the game. To make it a perfect match to WiF FE, Paul and Ullern had to do 1,5 months of calculations and statistical modeling, and thanks for their efforts, will will have in MWiF the same US Entry behavior as in WiF FE, so the same play balance in this regard. But any future adjustement will be a nightmare to make, as casual people are incapable of doing the analysis that Paul & Ullern have done, so we will be loosing control on any future adjustement possibility.


Let me develop.
The US Entry in WiF FE is such a delicate balance that it was tweaked twice in the WiF FE game's history. In 1996 was published the Final Edition of the game, with Countersheet (CD) 24 and 14 providing the US Entry markers.

In 2000, a new CS24 was published, it was the first tweaking. It was reported that US Entry was too variable. On this new CS24, 5 US Entry markers (3 of 1939 and 2 of 1940) had their value modified (some decreased some increased) :
1939
0 --> 1
3 --> 2
5 --> 4

1940
0 --> 1
0 --> 1
These modifications had the effect of tightening the distribution (std decreased) of chits around the average that decreased for 1939 and increased for 1940, thus giving a net result of less variability.


In 2007, a new CS24 was published, this was the second tweaking. On this new CS24, 7 US Entry markers (2 of 1939, 4 of 1940 and 1 of 1941) had their value modified (some decreased some increased) :
1939
0 --> 1
0 --> 1

1940
0 --> 1
0 --> 1
3 --> 2
4 --> 3

1941
3 --> 4
These modifications had the effect of again tightening the distribution of chits (std decreased) around the average that increased for 1939 did not change for 1940 and increased for 1941.


Manipulation of US Entry is manipulation to the play balance of the game.
Thanks to Paul and Ullern, we will have the chance to have MWiF behave very similary to WiF FE here.


But what will be done to MWiF if in the future it is discovered by the WiF FE developpment team that the US Entry needs another tweaking ?
US Entry tweaking is one of the central discussions amongst WiF players around the world (and it is so amongst people involved in the design of WiF FE), and embedding in the design of MWiF a factor that prevent any future modification only by specialists, is not desirable IMO.

I'm not saying that US Entry should be moddable at will by casual players, I'm just saying that we should keep in the design of MWiF a way of having it modified in the future so that it keeps matching WiF FE, to take advantage of any progress made in this section of the game.

Using a finite distribution was a work around for WIF. This is especially so since the neutrality chits were drawn from the same finite pool (e.g., the Nazi-Soviet pact is modelled by those major powers drawing chits with little US flags on them). Bringing in chits from the "future years" was also a work around and driven by the number of chits in the counter sheets. You might note that in each of the modifications you mentioned, additional chits were added; chits were not removed, nor were the same number of chits kept with modified values. The weakness of the finite pool is that the printing of chits costs money.
---
Only in exceptional cases do simulations use finite distributions for modelling. Usually that is due to a real world phenomenon which has a finite pool. For example, if you are modelling the selection of draft picks for a sport, there is a finite pool of players from which a team can draw.

What WIF is trying to model is the USA's likelihood of going to war. The change that introduced 'tension' into the design reflects that by its choice of the word tension. Actually, ADG introduced tension to the design prior to the modifications you listed. Unless I am mistaken, it was added to prevent the US player from never choosing any US Entry options and 'hoarding' his entry chits so he could declare war sooner. The addition of tension into the design forces the US to choose some US Entry options in order to eventually be capable of declaring war on the Axis powers.
---
Getting back to the current US Entry design, it is extremely hard to model. The US perception of geopolitical events was heavily influenced by events within the US. What appeared on the front page of the newspaper, or dominated the broadcast news (radio and movie reels in the 1930's and 1940's) most often concerned things like unemployment, government spending, taxes, natural disasters (e.g., hurricanes, floods, tornadoes), sports events, and celebrities (e.g., Hollywood). If something important happened elsewhere in the world during a slow news day, it would receive much more attention than if it happened during the World Series, or when a juicy scandal about a starlet was breaking news. My point is that any simulation of this is crude at best.

So, the goal for the US Entry design is not modelling reality precisely, but to provide the veri-similtude in game terms. What would be ideal, is if the game "plays out" US Entry roughly the way things would have happened if world events had been different. That is, if the decisions taken by the major powers had been different, as reflected by the WIF game events.
---
Infinite distributions are vastly preferred for simulations. They are a more robust tool. While you lament the amount of work that Paul and Nils had to spend to imitate the WIF FE finite pool, have you considered the effort that ADG had to go through to decide about the two tweaks you mention? I don't know how ADG made those decisions, but I do know that if we want to modify the outcome(s) of the MWIF model of the US Entry, it can be quickly evaluated using the same spreadsheets that Paul and Nils used.

The new design also permits having the distribution change during the year (e.g., every turn) if there is a need for finer control. We can also introduce feedback loops with virtually no effort. By feedback loops I mean that the increases in entry level could bias the distribution towards a distribution with higher values, or, similarly, decreases towards lower values. Those are both 'positive' feedback loops, where 'momentum' builds up or drops off. A negative feedback loop would be where increases in US Entry levels cause a distribution with lower values to be used.

My point is that neither of these refinements are available if you stay with using a finite pool.

Similar changes could be made to improve the modelling of the weather - though in MWIF I have made zero changes to how WIF FE models weather.
---
In conclusion, I believe that:
1 - An infinite distribution is a better simulation tool for this task.
2 - Paul and Nils have successfully defined infinite distributions that duplicate the net results of the WIF FE finite design.[&o][&o][&o]
3 - If a need/desire arises in the future for additional improvement, the infinite design permits finer tuning and provides more flexibility.
4 - And yes, I agree that future changes should not be made by the average player, but should be done by someone who has training in and/or experience with simulation models.

[Froonp]

ORIGINAL: Shannon V. OKeets
Using a finite distribution was a work around for WIF. This is especially so since the neutrality chits were drawn from the same finite pool (e.g., the Nazi-Soviet pact is modelled by those major powers drawing chits with little US flags on them). Bringing in chits from the "future years" was also a work around and driven by the number of chits in the counter sheets. You might note that in each of the modifications you mentioned, additional chits were added; chits were not removed, nor were the same number of chits kept with modified values. The weakness of the finite pool is that the printing of chits costs money.

No Chits were not added. The global number of chits stayed the same since WiF FE was published in 1996.
Exisiting chits (from CS24 always) had their value changed, as shown in my post. 0--> 1 means that a chit that was a 0 chit became a 1 chit.

For the rest, I understand what you say, it looks very good, a real achievement indeed, but in the end limiting the future modifications to only an elite who master statistics looks wrong to me. Maybe as an help the distribution Paul and Nills published should be mentionned, so that the numbers are known to the players.

[Shannon V. OKeets]

ORIGINAL: Froonp

ORIGINAL: Shannon V. OKeets
Using a finite distribution was a work around for WIF. This is especially so since the neutrality chits were drawn from the same finite pool (e.g., the Nazi-Soviet pact is modelled by those major powers drawing chits with little US flags on them). Bringing in chits from the "future years" was also a work around and driven by the number of chits in the counter sheets. You might note that in each of the modifications you mentioned, additional chits were added; chits were not removed, nor were the same number of chits kept with modified values. The weakness of the finite pool is that the printing of chits costs money.

No Chits were not added. The global number of chits stayed the same since WiF FE was published in 1996.
Exisiting chits (from CS24 always) had their value changed, as shown in my post. 0--> 1 means that a chit that was a 0 chit became a 1 chit.

For the rest, I understand what you say, it looks very good, a real achievement indeed, but in the end limiting the future modifications to only an elite who master statistics looks wrong to me. Maybe as an help the distribution Paul and Nills published should be mentionned, so that the numbers are known to the players.

I have already gotten permission from Paul and Nils to included their report as an included file when the game is released. The distribution itself will be included somewhere in the Players Manual (probably in the section on changes from WIF FE).

[paulderynck]

ORIGINAL: Froonp

For the rest, I understand what you say, it looks very good, a real achievement indeed, but in the end limiting the future modifications to only an elite who master statistics looks wrong to me. Maybe as an help the distribution Paul and Nills published should be mentionned, so that the numbers are known to the players.

The recommendation is in Post #1 above. The fact that it shows thousands of chits is not important because they really are more the "odds" of getting a chit of that value when dealing with an unlimited distribution. For example the odds of a zero chit at start and in 1939 is 50 out of 3017, or roughly 1 in 60, whereas the odds in the WIF boardgame are 1 in 30.

One thing that has not been talked about here much yet is the Standard Deviation of the accummulating US Entry total. With the proposed distribution it was far too great - which would have meant some really wild swings in US Entry (in both directions) would have been possible. To go with an unlimited pool and keep this Standard Deviation closer to the finite pool was definitely an objective. This was accomplished by reducing the quantities of the highest and lowest chits.

Not surprisingly [;)], I have a chart that illustrates this. Here are the average STDs of the accummulating US Entry totals of the three distributions for 5 of the Barb 41 strategies.




Note how the slope of the WIF plot increases at the start of each year but then rounds back down whereas the other two have a more constant slope. This is a direct result of the WIF one being a finite distribution.

[Froonp]

ORIGINAL: paulderynck
One thing that has not been talked about here much yet is the Standard Deviation of the accummulating US Entry total. With the proposed distribution it was far too great - which would have meant some really wild swings in US Entry (in both directions) would have been possible. To go with an unlimited pool and keep this Standard Deviation closer to the finite pool was definitely an objective. This was accomplished by reducing the quantities of the highest and lowest chits.

Not surprisingly, I have a chart that illustrates this. Here are the average STDs of the accummulating US Entry totals of the three distributions for 5 of the Barb 41 strategies.



Note how the slope of the WIF plot increases at the start of each year but then rounds back down whereas the other two have a more constant slope. This is a direct result of the WIF one being a finite distribution.

Also, the finite distribution seems to have a plateau to the Standard Deviation somewhere around 7.
Is something that your distribution can also mimic, or is this useless. After all, this only begins to be real different after M/J42.

[paulderynck]

ORIGINAL: Froonp
Also, the finite distribution seems to have a plateau to the Standard Deviation somewhere around 7.
Is something that your distribution can also mimic, or is this useless. After all, this only begins to be real different after M/J42.

This is true but to reduce the STD (standard deviation) of the recommendation any further I fear would result in almost eliminating the variability of the chit draws. You do want to keep some of that, so there is a "trade-off" (pardon the cliche). After all the iterations we tried, we ended up happy with the match of when targets were met. A couple examples are in Posts #4 and #5. Note in those how the original proposed "Inf" distribution has occurrences outside the bounds of the other two (in both directions) which are almost eliminated with the PN4 recommendation. Those were the "wild swings" due to STD that I was talking about.