1.2 Modelling Competitor + Strategy Probabilities in Trading Card Games

In the previous post, we discussed our decision to use Bayesian Inference as the preferred method to compute the win probabilities for different players and decks.

We will now expand on how Bayes’ Theorem can help us do this.

We’re going to take for granted that the probabilities here are examples only and that, given this demonstration, we’re also taking for granted that someway, somehow, all the probabilities have been provided to us.

In later parts of this project, we’ll tackle actually getting to the point where we can create such matchups with our own computed proabilities.

The Competitor + Strategy Matrix

Assume we have 8 Competitors (C₁…C₈) and 8 Strategies (S₁…S₈).

We might have a table that looks like this:

	C1	C2	C3	C4	C5	C6	C7	C8
S1	0.331	0.529	0.450	0.626	0.183	0.649	0.257	0.151
S2	0.507	0.271	0.537	0.200	0.258	0.513	0.119	0.330
S3	0.512	0.242	0.287	0.253	0.613	0.393	0.179	0.574
S4	0.677	0.540	0.694	0.610	0.406	0.138	0.209	0.395
S5	0.654	0.222	0.492	0.630	0.569	0.119	0.144	0.397
S6	0.297	0.544	0.534	0.262	0.243	0.432	0.696	0.352
S7	0.351	0.600	0.705	0.119	0.365	0.719	0.399	0.518
S8	0.415	0.342	0.477	0.110	0.205	0.432	0.493	0.134

This table tells us how probable a given Competitor, using a given Strategy is to win, in general. This would be a probability against a disembodied opponent: the probability isn’t conditioned on facing another person or deck. It’s most likely the long term average for the Competitors and their Strategies (wins/losses) expressed as a ratio.

Reading the table, above, we can see that Competitor 2 using Strategy 4 has a .540 probability of winning any game.

But what is this Competitor using this Strategy’s probability of winning against another Competitor and Strategy combination?

Computing Win Probabilities

Last time, we reviewed Bayes’ Theorem in the abstract:

[math] P(A|B)=\frac{P(A)P(B|A)}{P(B)} [/math]

We decided that the win probability for Player A against Player B was a conditional probability, or that this probability was conditioned on the circumstances of these two opponents facing one another.

It wouldn’t make sense for us simply to give a probability of Player A (or Player B) winning in isolation. This is what our table, above, does.

It doesn’t, however, give us a probability of a Competitor + Strategy combination against another Competitor + Strategy combination, which is what we want.

Let’s take the case of Competitor 3 using Strategy 5 (C₃+S₅) playing against Competitor 7 using Strategy 2 (C7+S2).

How do we calculate the probability that C₃+S₅ win a match against C7+S2?

We set up our equation thusly:

[math] P(C3+S5|C7+S3)=\frac{P(C3+S5)P(C7+ S3|C3+S5)}{C7+ S5} [/math]

Let’s examine each of the variables in turn.

P(C₃+S₅ | C₇ + S₃)

This is the win probability of C₃+S₅ against C₇+S₃, which is what we’re looking for.

We won’t know this until the end!

P(C₃+S₅)

Referring to our table above, we find that the probability of C₃+S₅ winning any match is 0.492.

Thus, the value of P(C₃+S₅) is 0.492 in our formula.

P(C₇+S₃|C₃+S₅)

This one is a little trickier. This is the probability that C7+S3 win against C₃+S₅.

And if we think about it, is the opposite of what we’re looking for, P(C₃+S₅|C₇+S₃)!

If we knew this now, we’d be done, because to find the other probability, we simply subtract it from 1. This is something we’ll return to shortly.

For now, let’s leave this as a mystery.

P(C₇+S₃)

This is simply the probability that C7+S3 win any match.

Consulting our table, we see that this is 0.179.

So we’ll use that.

Missing Information

So far, our revised formula looks like this:

[math] P(C3+S5| C7+ S3)= \frac{0.492*P(C7+ S3|C3+S5)}{P(0.179)}[/math]

But what about that pesky P(C₇+S₃|C₃+S₅)?

If we could calculate that, we’d know P(C₃+S₅ | C₇+ S₃), our desired answer, too.

Let’s find an easier way to make this calculation.

Using Excel and Basic Algebra

Let’s take another look at our Competitor and Strategy table:

We know that C₃+S₅ has an overall win probability of 0.492 and that C₇+S₃ has an overall win probability of 0.179.

To make our life easier, we will constrain the probability space to the universe that contains only these two match ups. That is, to say, that 1 (or 100%) will be the combined total probabilities for these two Competitors + Strategies.

We can think of this as a ratio:

[math] Win Probability= \frac{P(C3+C5)}{P(C3+C5) + P(C7+ S3)}[/math]

This pits the probability of C₃+C₅ against the combined probability of C₃+C₅ and C₇+ S₃.

We get the following:

[math]  \frac{0.492}{0.492+0.179}=0.733[/math]

This works because the probability that C₃+C₅ win any match is a fraction of the combined probabilities that C₃+C₅ win any match and C₇+S₃ win any match.

In fact, we can put all of this back into Bayes’ Theorem to prove our case.

Remember, that if the P(C₃+C₅|P(C₇ + S₃)) = 0.733, then the probability that the other player win is exactly 1 – this amount, or P(C₇₌ + S₃|C₃+C₅) = 1 – 0.733 = 0.267.

This is because all probabilities must sum to 1 (absolute certainty). All potential outcomes represent the whole of possible outcomes. 

[math]P(C3+S5| C7+ S3)=\frac{0.492*0.267}{0.179} =0.733[/math]

All we have to do is divide C_n+S_n by C_n+S_n combined with C_k+S_k.

This is much easier.

What we’ve done here is normalize the probabilities of the two competitors such that their conditional probabilities sum to 1, like this:

The Expanded C_n+S_n Matrix

To carry this theory into action, we pit each and every combination of Competitors + Strategies (C_n+S_n) against each other in the same way as we did above for C₃+C₅ against C₇+S₃.

Our previous win probability table has 8 rows and 8 columns. This one has 64 rows and 64 columns.

Wins? ↓	C1+S1	C1+S2	C1+S3	C1+S4	C1+S5	C1+S6	C1+S7	C1+S8	C2+S1	C2+S2	C2+S3	C2+S4	C2+S5	C2+S6	C2+S7	C2+S8	C3+S1	C3+S2	C3+S3	C3+S4	C3+S5	C3+S6	C3+S7	C3+S8	C4+S1	C4+S2	C4+S3	C4+S4	C4+S5	C4+S6	C4+S7	C4+S8	C5+S1	C5+S2	C5+S3	C5+S4	C5+S5	C5+S6	C5+S7	C5+S8	C6+S1	C6+S2	C6+S3	C6+S4	C6+S5	C6+S6	C6+S7	C6+S8	C7+S1	C7+S2	C7+S3	C7+S4	C7+S5	C7+S6	C7+S7	C7+S8	C8+S1	C8+S2	C8+S3	C8+S4	C8+S5	C8+S6	C8+S7	C8+S8
C1+S1	0.5	0.394988067	0.392645314	0.328373016	0.336040609	0.527070064	0.485337243	0.443699732	0.384883721	0.549833887	0.577661431	0.380022962	0.598553345	0.378285714	0.355531686	0.491827637	0.423815621	0.381336406	0.535598706	0.322926829	0.40218712	0.38265896	0.319498069	0.409653465	0.345872518	0.623352166	0.566780822	0.351753454	0.344432882	0.558178752	0.735555556	0.750566893	0.643968872	0.56196944	0.350635593	0.449118046	0.367777778	0.576655052	0.475574713	0.617537313	0.337755102	0.392180095	0.45718232	0.70575693	0.735555556	0.433813893	0.315238095	0.433813893	0.56292517	0.735555556	0.649019608	0.612962963	0.696842105	0.322297955	0.453424658	0.401699029	0.686721992	0.50075643	0.365745856	0.455922865	0.45467033	0.484626647	0.389870436	0.711827957
C1+S2	0.605011933	0.5	0.497546614	0.428209459	0.436692506	0.630597015	0.590909091	0.54989154	0.489382239	0.651670951	0.676902537	0.484240688	0.695473251	0.482397716	0.45799458	0.597173145	0.529780564	0.485632184	0.638539043	0.42214821	0.507507508	0.4870317	0.418316832	0.515243902	0.447484554	0.717114569	0.667105263	0.45389436	0.44591029	0.659297789	0.809904153	0.82171799	0.734782609	0.662745098	0.452678571	0.555312158	0.471189591	0.676	0.581422018	0.712078652	0.438581315	0.497058824	0.563333333	0.786046512	0.809904153	0.539936102	0.413539967	0.539936102	0.663612565	0.809904153	0.739067055	0.708100559	0.778801843	0.421446384	0.559602649	0.507	0.770516717	0.605734767	0.469010176	0.562084257	0.560840708	0.590221187	0.494634146	0.790951638
C1+S3	0.607354686	0.502453386	0.5	0.430613961	0.439108062	0.632880099	0.593279258	0.55231931	0.491834774	0.653895275	0.679045093	0.486692015	0.697547684	0.484848485	0.460431655	0.599531616	0.532224532	0.488083889	0.640801001	0.424543947	0.509960159	0.489483748	0.420706656	0.517694641	0.449912127	0.719101124	0.669281046	0.456327986	0.448336252	0.661498708	0.81141046	0.823151125	0.736690647	0.664935065	0.455111111	0.557734205	0.473635523	0.678145695	0.583808438	0.714086471	0.440999139	0.499512195	0.565745856	0.787692308	0.81141046	0.542372881	0.415922015	0.542372881	0.66579974	0.81141046	0.740955137	0.710124827	0.780487805	0.42384106	0.562019759	0.509452736	0.77224736	0.60807601	0.47145488	0.564498346	0.563256326	0.592592593	0.497087379	0.792569659
C1+S4	0.671626984	0.571790541	0.569386039	0.5	0.50864012	0.695071869	0.658560311	0.61996337	0.561359867	0.714135021	0.736670294	0.556285949	0.753058954	0.554463554	0.530148786	0.66437684	0.600709849	0.557660626	0.702282158	0.493800146	0.579127459	0.559042114	0.489869754	0.586655113	0.519570223	0.771949829	0.727956989	0.526029526	0.517980107	0.720979766	0.850502513	0.860228717	0.787209302	0.724064171	0.524806202	0.62511542	0.543338684	0.735869565	0.649712092	0.767573696	0.510558069	0.568907563	0.63271028	0.830674847	0.850502513	0.610459874	0.48495702	0.610459874	0.7248394	0.850502513	0.79088785	0.764108352	0.824604141	0.493080845	0.629182156	0.578632479	0.81763285	0.672293942	0.541167066	0.631529851	0.630353818	0.657920311	0.566527197	0.834771887
C1+S5	0.663959391	0.563307494	0.560891938	0.49135988	0.5	0.687697161	0.650746269	0.611786717	0.552831784	0.707027027	0.729910714	0.547738693	0.746575342	0.54590985	0.5215311	0.656626506	0.592391304	0.549118388	0.695005313	0.485163205	0.570680628	0.550505051	0.481236203	0.578249337	0.5109375	0.765807963	0.721058434	0.517405063	0.509345794	0.713973799	0.846054334	0.856020942	0.781362007	0.717105263	0.516179953	0.616981132	0.534750613	0.72909699	0.641805692	0.761350407	0.501918649	0.560411311	0.624641834	0.825757576	0.846054334	0.602209945	0.476329206	0.602209945	0.717892426	0.846054334	0.785114046	0.757821553	0.819548872	0.484444444	0.621082621	0.570183086	0.81242236	0.664634146	0.53257329	0.623450906	0.62226451	0.650099404	0.558020478	0.829949239
C1+S6	0.472929936	0.369402985	0.367119901	0.304928131	0.312302839	0.5	0.458333333	0.417134831	0.359564165	0.522887324	0.551020408	0.35483871	0.572254335	0.353151011	0.331103679	0.464788732	0.397590361	0.356115108	0.508561644	0.299697275	0.376425856	0.357400722	0.296407186	0.38372093	0.321776815	0.597585513	0.54	0.327453142	0.32038835	0.531305903	0.713942308	0.72972973	0.61875	0.535135135	0.326373626	0.422475107	0.34295612	0.55	0.448640483	0.591633466	0.313953488	0.366666667	0.430434783	0.682758621	0.713942308	0.407407407	0.292322835	0.407407407	0.536101083	0.713942308	0.62394958	0.586956522	0.673469388	0.299093656	0.426724138	0.375949367	0.662946429	0.473684211	0.340987371	0.429190751	0.42795389	0.457627119	0.364417178	0.689095128
C1+S7	0.514662757	0.409090909	0.406720742	0.341439689	0.349253731	0.541666667	0.5	0.458224543	0.398863636	0.564308682	0.591905565	0.393939394	0.612565445	0.392178771	0.369085174	0.506493506	0.438202247	0.39527027	0.55015674	0.335885167	0.416370107	0.396610169	0.332386364	0.423913043	0.35926305	0.637023593	0.581125828	0.365244537	0.357798165	0.572593801	0.746808511	0.761388286	0.657303371	0.57635468	0.364107884	0.463672391	0.381521739	0.590909091	0.490223464	0.631294964	0.351	0.40625	0.471774194	0.717791411	0.746808511	0.448275862	0.328037383	0.448275862	0.577302632	0.746808511	0.662264151	0.626785714	0.709090909	0.335243553	0.468	0.415876777	0.699203187	0.515418502	0.379459459	0.470509383	0.469251337	0.499288762	0.403912543	0.72371134
C1+S8	0.556300268	0.45010846	0.44768069	0.38003663	0.388213283	0.582865169	0.541775457	0.5	0.439618644	0.604956268	0.631659056	0.434554974	0.651491366	0.43274244	0.408866995	0.548216645	0.479768786	0.43592437	0.591168091	0.374211001	0.45755237	0.437302424	0.370535714	0.465246637	0.398655139	0.674796748	0.621257485	0.404878049	0.397129187	0.612998523	0.777153558	0.79047619	0.693979933	0.616641902	0.403696498	0.505481121	0.421747967	0.630699088	0.532051282	0.669354839	0.390037594	0.447198276	0.513613861	0.75045208	0.777153558	0.489964581	0.365961199	0.489964581	0.617559524	0.777153558	0.698653199	0.665064103	0.742397138	0.373537354	0.50982801	0.457048458	0.733215548	0.55704698	0.419615774	0.512345679	0.511083744	0.5410691	0.444801715	0.755919854
C2+S1	0.615116279	0.510617761	0.508165226	0.438640133	0.447168216	0.640435835	0.601136364	0.560381356	0.5	0.66125	0.68612192	0.494855005	0.704394141	0.493010252	0.468556244	0.607347876	0.540347293	0.496247655	0.648284314	0.432542927	0.518119491	0.497648166	0.428687196	0.52584493	0.458008658	0.725651578	0.676470588	0.464442493	0.456427955	0.668773704	0.816358025	0.827856025	0.742977528	0.672172808	0.463222417	0.565775401	0.481785064	0.685233161	0.591722595	0.720708447	0.449066214	0.507677543	0.573752711	0.793103448	0.816358025	0.550468262	0.423878205	0.550468262	0.67302799	0.816358025	0.747175141	0.716802168	0.786032689	0.431836735	0.570043103	0.517612524	0.777941176	0.615832363	0.479601088	0.572510823	0.571274298	0.60045403	0.505253104	0.797888386
C2+S2	0.450166113	0.348329049	0.346104725	0.285864979	0.292972973	0.477112676	0.435691318	0.395043732	0.33875	0.5	0.528265107	0.334155364	0.54969574	0.332515337	0.311136625	0.442088091	0.375866852	0.33539604	0.485663082	0.280829016	0.355176933	0.336645963	0.277663934	0.362299465	0.302118172	0.57537155	0.517175573	0.307604994	0.300776915	0.508442777	0.694871795	0.711286089	0.5969163	0.512287335	0.306561086	0.400295421	0.322619048	0.527237354	0.426100629	0.569327731	0.294565217	0.345663265	0.40813253	0.662591687	0.694871795	0.385490754	0.273737374	0.385490754	0.513257576	0.694871795	0.602222222	0.564583333	0.653012048	0.28024819	0.404477612	0.354712042	0.642180095	0.450915141	0.320710059	0.406906907	0.405688623	0.434991974	0.34347275	0.669135802
C2+S3	0.422338569	0.323097463	0.320954907	0.263329706	0.270089286	0.448979592	0.408094435	0.368340944	0.31387808	0.471734893	0.5	0.309462916	0.521551724	0.307888041	0.287410926	0.414383562	0.349710983	0.310654685	0.457466919	0.258547009	0.329700272	0.31185567	0.255543823	0.336578581	0.278801843	0.547511312	0.488888889	0.284037559	0.277522936	0.48015873	0.670360111	0.6875	0.569411765	0.484	0.283040936	0.37345679	0.298397041	0.498969072	0.398682043	0.541387025	0.271604938	0.320529801	0.381102362	0.636842105	0.670360111	0.359050445	0.25182102	0.359050445	0.48496994	0.670360111	0.574821853	0.536585366	0.626943005	0.257995736	0.377535101	0.329251701	0.615776081	0.423076923	0.296568627	0.379905808	0.378716745	0.407407407	0.318421053	0.643617021
C2+S4	0.619977038	0.515759312	0.513307985	0.443714051	0.452261307	0.64516129	0.606060606	0.565445026	0.505144995	0.665844636	0.690537084	0.5	0.708661417	0.498154982	0.473684211	0.612244898	0.545454545	0.501392758	0.652962515	0.437601297	0.523255814	0.502793296	0.43373494	0.530973451	0.463121784	0.72972973	0.680958386	0.469565217	0.461538462	0.673316708	0.819423369	0.830769231	0.746887967	0.676691729	0.468343452	0.570824524	0.486925158	0.689655172	0.596685083	0.724832215	0.454163162	0.512820513	0.578778135	0.796460177	0.819423369	0.555555556	0.428911835	0.555555556	0.677540778	0.819423369	0.751043115	0.720961282	0.789473684	0.436893204	0.575079872	0.522749274	0.781476122	0.620689655	0.484739677	0.577540107	0.576307364	0.605381166	0.510396975	0.801186944
C2+S5	0.401446655	0.304526749	0.302452316	0.246941046	0.253424658	0.427745665	0.387434555	0.348508634	0.295605859	0.45030426	0.478448276	0.291338583	0.5	0.289817232	0.270072993	0.393617021	0.330357143	0.292490119	0.436149312	0.242358079	0.31092437	0.293650794	0.239482201	0.317596567	0.261792453	0.526066351	0.467368421	0.266826923	0.26056338	0.458677686	0.651026393	0.668674699	0.548148148	0.4625	0.265868263	0.353503185	0.280657396	0.477419355	0.378194208	0.519906323	0.254879449	0.302040816	0.36097561	0.616666667	0.651026393	0.339449541	0.235919235	0.339449541	0.463465553	0.651026393	0.55361596	0.515081206	0.606557377	0.241830065	0.357487923	0.31048951	0.595174263	0.402173913	0.278894472	0.359805511	0.358642973	0.386759582	0.3	0.623595506
C2+S6	0.621714286	0.517602284	0.515151515	0.445536446	0.45409015	0.646848989	0.607821229	0.56725756	0.506989748	0.667484663	0.692111959	0.501845018	0.710182768	0.5	0.475524476	0.613995485	0.547283702	0.503237743	0.654632972	0.439418417	0.525096525	0.504638219	0.435548439	0.53281097	0.464957265	0.731182796	0.682559598	0.471403813	0.463373083	0.674937965	0.820512821	0.831804281	0.748280605	0.678304239	0.470181504	0.572631579	0.488769093	0.691232529	0.598459846	0.726301736	0.455993294	0.514664144	0.580576307	0.797653959	0.820512821	0.557377049	0.430720507	0.557377049	0.679151061	0.820512821	0.75242047	0.722443559	0.790697674	0.438709677	0.576882291	0.524590164	0.782733813	0.622425629	0.486583184	0.579339723	0.578108395	0.607142857	0.512241055	0.802359882
C2+S7	0.644468314	0.54200542	0.539568345	0.469851214	0.4784689	0.668896321	0.630914826	0.591133005	0.531443756	0.688863375	0.712589074	0.526315789	0.729927007	0.524475524	0.5	0.636942675	0.571428571	0.527704485	0.67643743	0.463678516	0.549450549	0.529100529	0.459770115	0.557103064	0.489396411	0.75	0.703399766	0.495867769	0.487804878	0.696055684	0.83449235	0.845070423	0.766283525	0.699300699	0.494641385	0.596421471	0.513259196	0.711743772	0.621761658	0.745341615	0.480384307	0.539083558	0.604229607	0.81300813	0.83449235	0.581395349	0.454890068	0.581395349	0.700116686	0.83449235	0.770218228	0.741656366	0.806451613	0.462962963	0.600600601	0.54894785	0.798934754	0.64516129	0.511073254	0.603015075	0.601805416	0.630252101	0.53667263	0.817438692
C2+S8	0.508172363	0.402826855	0.400468384	0.33562316	0.343373494	0.535211268	0.493506494	0.451783355	0.392652124	0.557911909	0.585616438	0.387755102	0.606382979	0.386004515	0.363057325	0.5	0.431818182	0.389078498	0.543720191	0.33011583	0.410071942	0.390410959	0.326647564	0.417582418	0.353305785	0.63099631	0.574789916	0.359243697	0.351851852	0.566225166	0.74186551	0.756637168	0.651428571	0.57	0.358115183	0.457219251	0.375411636	0.584615385	0.483734088	0.625228519	0.345105954	0.4	0.465306122	0.7125	0.74186551	0.441860465	0.322337418	0.441860465	0.570951586	0.74186551	0.656429942	0.620689655	0.703703704	0.329479769	0.461538462	0.409580838	0.693711968	0.508928571	0.373362445	0.464043419	0.462787551	0.492795389	0.397674419	0.718487395
C3+S1	0.576184379	0.470219436	0.467775468	0.399290151	0.407608696	0.602409639	0.561797753	0.520231214	0.459652707	0.624133148	0.650289017	0.454545455	0.669642857	0.452716298	0.428571429	0.568181818	0.5	0.455927052	0.610583446	0.393356643	0.477707006	0.457317073	0.38961039	0.485436893	0.418215613	0.692307692	0.640113798	0.424528302	0.416666667	0.632022472	0.79086116	0.803571429	0.710900474	0.63559322	0.423330198	0.525700935	0.441609421	0.649350649	0.552147239	0.687022901	0.409463148	0.46728972	0.533807829	0.765306122	0.79086116	0.510204082	0.384944397	0.510204082	0.636492221	0.79086116	0.715421304	0.682852807	0.757575758	0.392670157	0.530035336	0.477200424	0.74875208	0.576923077	0.439453125	0.532544379	0.531286895	0.561097257	0.464876033	0.770547945
C3+S2	0.618663594	0.514367816	0.511916111	0.442339374	0.450881612	0.643884892	0.60472973	0.56407563	0.503752345	0.66460396	0.689345315	0.498607242	0.707509881	0.496762257	0.472295515	0.610921502	0.544072948	0.5	0.651699029	0.436230707	0.521865889	0.50140056	0.43236715	0.529585799	0.461736887	0.728629579	0.679746835	0.468177855	0.460154242	0.672090113	0.818597561	0.829984544	0.745833333	0.675471698	0.466956522	0.569459173	0.485533454	0.688461538	0.595343681	0.723719677	0.452782462	0.511428571	0.577419355	0.795555556	0.818597561	0.554179567	0.427547771	0.554179567	0.676322418	0.818597561	0.75	0.719839142	0.788546256	0.435523114	0.573717949	0.521359223	0.780523256	0.619377163	0.483348335	0.576180258	0.574946467	0.604049494	0.509004739	0.800298063
C3+S3	0.464401294	0.361460957	0.359198999	0.297717842	0.304994687	0.491438356	0.44984326	0.408831909	0.351715686	0.514336918	0.542533081	0.347037485	0.563850688	0.345367028	0.32356257	0.456279809	0.389416554	0.348300971	0.5	0.292558614	0.368421053	0.349573691	0.289314516	0.37565445	0.314348302	0.589322382	0.531481481	0.319955407	0.312977099	0.52276867	0.706896552	0.722921914	0.610638298	0.526605505	0.318888889	0.414141414	0.335280374	0.541509434	0.440184049	0.583333333	0.306623932	0.35875	0.422058824	0.675294118	0.706896552	0.399165508	0.28528827	0.399165508	0.527573529	0.706896552	0.615879828	0.578629032	0.665893271	0.291963377	0.418367347	0.367948718	0.655251142	0.465153971	0.333333333	0.420821114	0.419590643	0.44913928	0.356521739	0.681710214
C3+S4	0.677073171	0.57785179	0.575456053	0.506199854	0.514836795	0.700302725	0.664114833	0.625788999	0.567457073	0.719170984	0.741452991	0.562398703	0.757641921	0.560581583	0.536321484	0.66988417	0.606643357	0.563769293	0.707441386	0.5	0.585160202	0.56514658	0.49606862	0.59265585	0.525757576	0.776286353	0.732840549	0.532208589	0.524169184	0.725941423	0.853628536	0.86318408	0.791334094	0.728991597	0.530986993	0.630909091	0.549485352	0.740661686	0.655335222	0.771968854	0.516753537	0.574979287	0.638454462	0.834134615	0.853628536	0.61634103	0.491153574	0.61634103	0.729758149	0.853628536	0.794959908	0.76854928	0.828162291	0.499280576	0.63494968	0.584667228	0.821301775	0.677734375	0.547318612	0.63728191	0.636113657	0.663479924	0.572607261	0.838164251
C3+S5	0.59781288	0.492492492	0.490039841	0.420872541	0.429319372	0.623574144	0.583629893	0.54244763	0.481880509	0.644823067	0.670299728	0.476744186	0.68907563	0.474903475	0.450549451	0.589928058	0.522292994	0.478134111	0.631578947	0.414839798	0.5	0.479532164	0.411027569	0.507739938	0.440071556	0.710982659	0.660402685	0.44646098	0.438502674	0.652519894	0.805237316	0.817275748	0.728888889	0.656	0.445248869	0.547884187	0.463713478	0.669387755	0.574095683	0.705882353	0.431200701	0.489552239	0.555932203	0.780952381	0.805237316	0.532467532	0.406275805	0.532467532	0.656875834	0.805237316	0.733233979	0.701854494	0.773584906	0.414141414	0.552188552	0.499492386	0.765163297	0.598540146	0.461538462	0.554678692	0.553430821	0.582938389	0.487128713	0.785942492
C3+S6	0.61734104	0.5129683	0.510516252	0.440957886	0.449494949	0.642599278	0.603389831	0.562697576	0.502351834	0.663354037	0.68814433	0.497206704	0.706349206	0.495361781	0.470899471	0.609589041	0.542682927	0.49859944	0.650426309	0.43485342	0.520467836	0.5	0.430992736	0.528189911	0.460344828	0.727520436	0.678526048	0.466783217	0.458762887	0.670854271	0.817764165	0.829192547	0.744769874	0.674242424	0.465562337	0.568085106	0.48413418	0.687258687	0.593993326	0.722598106	0.451394759	0.510028653	0.57605178	0.794642857	0.817764165	0.552795031	0.426177175	0.552795031	0.675094817	0.817764165	0.748948107	0.718707941	0.787610619	0.434146341	0.572347267	0.519961052	0.779562044	0.618055556	0.481949458	0.574811625	0.573576799	0.602708804	0.507604563	0.799401198
C3+S7	0.680501931	0.581683168	0.579293344	0.510130246	0.518763797	0.703592814	0.667613636	0.629464286	0.571312804	0.722336066	0.744456177	0.56626506	0.760517799	0.564451561	0.540229885	0.673352436	0.61038961	0.56763285	0.710685484	0.50393138	0.588972431	0.569007264	0.5	0.596446701	0.529676935	0.779005525	0.735908142	0.536121673	0.528089888	0.729058945	0.855582524	0.865030675	0.793918919	0.732087227	0.534901366	0.634563456	0.553375196	0.743670886	0.658878505	0.774725275	0.520679468	0.578817734	0.642076503	0.836298932	0.855582524	0.62005277	0.49508427	0.62005277	0.732848233	0.855582524	0.797511312	0.771334792	0.830388693	0.503211991	0.638586957	0.588480801	0.823598131	0.68115942	0.551211884	0.640909091	0.639745917	0.666982025	0.576451349	0.840286055
C3+S8	0.590346535	0.484756098	0.482305359	0.413344887	0.421750663	0.61627907	0.576086957	0.534753363	0.47415507	0.637700535	0.663421419	0.469026549	0.682403433	0.46718903	0.442896936	0.582417582	0.514563107	0.470414201	0.62434555	0.40734415	0.492260062	0.471810089	0.403553299	0.5	0.432456936	0.704579025	0.653424658	0.438822447	0.430894309	0.645466847	0.80033557	0.812606474	0.722727273	0.648979592	0.437614679	0.540203851	0.456022945	0.6625	0.566508314	0.69941349	0.423623446	0.481818182	0.548275862	0.775609756	0.80033557	0.524752475	0.398829431	0.524752475	0.64986376	0.80033557	0.727134146	0.695335277	0.768115942	0.406649616	0.544520548	0.491752577	0.75955414	0.591078067	0.453853473	0.547018349	0.54576659	0.575392039	0.479396985	0.780687398
C4+S1	0.654127482	0.552515446	0.550087873	0.480429777	0.4890625	0.218637993	0.64073695	0.601344861	0.541991342	0.697881828	0.721198157	0.536878216	0.738207547	0.535042735	0.510603589	0.646694215	0.581784387	0.538263113	0.685651698	0.474242424	0.559928444	0.539655172	0.470323065	0.567543064	0.5	0.757869249	0.712172924	0.506472492	0.498407643	0.704954955	0.840268456	0.850543478	0.773794808	0.708144796	0.505246166	0.606589147	0.523849372	0.720368239	0.631685166	0.753309266	0.490980392	0.549604917	0.614327772	0.819371728	0.840268456	0.59168242	0.465427509	0.59168242	0.708946772	0.840268456	0.777639752	0.749700599	0.812987013	0.473524962	0.610731707	0.559428061	0.805662806	0.654811715	0.521666667	0.613124388	0.611925709	0.6400818	0.547202797	0.823684211
C4+S2	0.376647834	0.282885431	0.280898876	0.228050171	0.234192037	0.282894737	0.362976407	0.325203252	0.274348422	0.42462845	0.452488688	0.27027027	0.473933649	0.268817204	0.25	0.36900369	0.307692308	0.271370421	0.410677618	0.223713647	0.289017341	0.272479564	0.220994475	0.295420975	0.242130751	0.5	0.441501104	0.24691358	0.240963855	0.432900433	0.626959248	0.64516129	0.522193211	0.436681223	0.24600246	0.330033003	0.260078023	0.451467269	0.353982301	0.49382716	0.23557126	0.280504909	0.337268128	0.591715976	0.626959248	0.316455696	0.217627856	0.316455696	0.437636761	0.626959248	0.527704485	0.488997555	0.581395349	0.223214286	0.333889816	0.288600289	0.56980057	0.377358491	0.258397933	0.336134454	0.335008375	0.362318841	0.278551532	0.598802395
C4+S3	0.433219178	0.332894737	0.330718954	0.272043011	0.278941566	0.483820047	0.418874172	0.378742515	0.323529412	0.482824427	0.511111111	0.319041614	0.532631579	0.317440402	0.296600234	0.425210084	0.359886202	0.320253165	0.468518519	0.267159451	0.339597315	0.321473952	0.264091858	0.346575342	0.287827076	0.558498896	0.5	0.293163384	0.286523216	0.491262136	0.680107527	0.696969697	0.580275229	0.495107632	0.292147806	0.383915023	0.307785888	0.510080645	0.409385113	0.552401747	0.280487805	0.330287206	0.391640867	0.647058824	0.680107527	0.369343066	0.260288066	0.369343066	0.496078431	0.680107527	0.585648148	0.547619048	0.637279597	0.266596417	0.38803681	0.339142091	0.626237624	0.433962264	0.30592503	0.390432099	0.389230769	0.418181818	0.328145266	0.65374677
C4+S4	0.648246546	0.54610564	0.543672014	0.473970474	0.482594937	0.383018868	0.634755463	0.595121951	0.535557507	0.692395006	0.715962441	0.530434783	0.733173077	0.528596187	0.504132231	0.640756303	0.575471698	0.531822145	0.680044593	0.467791411	0.55353902	0.533216783	0.463878327	0.561177553	0.493527508	0.75308642	0.706836616	0.5	0.491935484	0.699541284	0.836762689	0.847222222	0.769230769	0.702764977	0.498773508	0.600393701	0.517387617	0.715123095	0.625641026	0.748466258	0.484511517	0.54318789	0.608175474	0.815508021	0.836762689	0.585412668	0.458991723	0.585412668	0.703575548	0.836762689	0.773130545	0.744810745	0.809018568	0.467075038	0.604558969	0.553037171	0.801576873	0.64893617	0.515202703	0.606965174	0.605759682	0.634095634	0.540780142	0.819892473
C4+S5	0.655567118	0.55408971	0.551663748	0.482019893	0.490654206	0.465249387	0.642201835	0.602870813	0.543572045	0.699223085	0.722477064	0.538461538	0.73943662	0.536626917	0.512195122	0.648148148	0.583333333	0.539845758	0.687022901	0.475830816	0.561497326	0.541237113	0.471910112	0.569105691	0.501592357	0.759036145	0.713476784	0.508064516	0.5	0.706278027	0.841121495	0.851351351	0.774907749	0.709459459	0.506838294	0.608108108	0.525437865	0.721649485	0.633165829	0.754491018	0.492572322	0.551181102	0.615835777	0.8203125	0.841121495	0.593220339	0.467012602	0.593220339	0.710259301	0.841121495	0.778739184	0.750893921	0.813953488	0.475113122	0.612244898	0.560997329	0.806658131	0.65625	0.523255814	0.614634146	0.613437196	0.641547862	0.548780488	0.82460733
C4+S6	0.441821248	0.340702211	0.338501292	0.279020234	0.286026201	0.27090301	0.427406199	0.387001477	0.331226296	0.491557223	0.51984127	0.326683292	0.541322314	0.325062035	0.303944316	0.433774834	0.367977528	0.327909887	0.47723133	0.274058577	0.347480106	0.329145729	0.270941055	0.354533153	0.295045045	0.567099567	0.508737864	0.300458716	0.293721973	0.5	0.687664042	0.704301075	0.588764045	0.503846154	0.299428571	0.392215569	0.315282792	0.518811881	0.417862839	0.561027837	0.287596048	0.338064516	0.4	0.655	0.687664042	0.377521614	0.267074414	0.377521614	0.504816956	0.687664042	0.594104308	0.55626327	0.645320197	0.27348643	0.396369138	0.347019868	0.634382567	0.442567568	0.313397129	0.398782344	0.397572079	0.426710098	0.335897436	0.661616162
C4+S7	0.264444444	0.190095847	0.18858954	0.149497487	0.153945666	0.358194308	0.253191489	0.222846442	0.183641975	0.305128205	0.329639889	0.180576631	0.348973607	0.179487179	0.16550765	0.25813449	0.20913884	0.181402439	0.293103448	0.146371464	0.194762684	0.182235835	0.144417476	0.19966443	0.159731544	0.373040752	0.319892473	0.163237311	0.158878505	0.312335958	0.5	0.519650655	0.394039735	0.315649867	0.162568306	0.226666667	0.172965116	0.328729282	0.245867769	0.367283951	0.154947917	0.188291139	0.232421875	0.463035019	0.5	0.215970962	0.142004773	0.215970962	0.316489362	0.5	0.399328859	0.362804878	0.452471483	0.14601227	0.22972973	0.194444444	0.440740741	0.265033408	0.171717172	0.23151751	0.230620155	0.252653928	0.186813187	0.470355731
C4+S8	0.249433107	0.17828201	0.176848875	0.139771283	0.143979058	0.238649593	0.238611714	0.20952381	0.172143975	0.288713911	0.3125	0.169230769	0.331325301	0.168195719	0.154929577	0.243362832	0.196428571	0.170015456	0.277078086	0.13681592	0.182724252	0.170807453	0.134969325	0.187393526	0.149456522	0.35483871	0.303030303	0.152777778	0.148648649	0.295698925	0.480349345	0.5	0.375426621	0.298913043	0.152143845	0.213178295	0.162002946	0.311614731	0.231578947	0.349206349	0.144927536	0.176565008	0.218687873	0.443548387	0.480349345	0.20295203	0.132689988	0.20295203	0.29972752	0.480349345	0.380622837	0.344827586	0.433070866	0.136476427	0.21611002	0.182421227	0.421455939	0.25	0.160818713	0.217821782	0.216962525	0.238095238	0.175159236	0.450819672
C5+S1	0.356031128	0.265217391	0.263309353	0.212790698	0.218637993	0.218637993	0.342696629	0.306020067	0.257022472	0.4030837	0.430588235	0.253112033	0.451851852	0.251719395	0.233716475	0.348571429	0.289099526	0.254166667	0.389361702	0.208665906	0.271111111	0.255230126	0.206081081	0.277272727	0.226205192	0.477806789	0.419724771	0.230769231	0.225092251	0.411235955	0.605960265	0.624573379	0.5	0.414965986	0.229899497	0.310696095	0.243351064	0.429577465	0.333941606	0.471649485	0.219951923	0.262931034	0.317708333	0.570093458	0.605960265	0.297560976	0.202882483	0.297560976	0.415909091	0.605960265	0.505524862	0.466836735	0.559633028	0.208191126	0.31443299	0.270710059	0.547904192	0.356725146	0.241743725	0.316608997	0.315517241	0.342056075	0.261055635	0.577287066
C5+S2	0.43803056	0.337254902	0.335064935	0.275935829	0.282894737	0.282894737	0.42364532	0.383358098	0.327827192	0.487712665	0.516	0.323308271	0.5375	0.321695761	0.300699301	0.43	0.36440678	0.324528302	0.473394495	0.271008403	0.344	0.325757576	0.267912773	0.351020408	0.291855204	0.563318777	0.504892368	0.297235023	0.290540541	0.496153846	0.684350133	0.701086957	0.585034014	0.5	0.296211251	0.388554217	0.311970979	0.51497006	0.414125201	0.557235421	0.284454245	0.33463035	0.396313364	0.651515152	0.684350133	0.373913043	0.264073695	0.373913043	0.500970874	0.684350133	0.590389016	0.552462527	0.641791045	0.270440252	0.392694064	0.343541944	0.630806846	0.43877551	0.310096154	0.395099541	0.39389313	0.42295082	0.332474227	0.658163265
C5+S3	0.649364407	0.547321429	0.544888889	0.475193798	0.483820047	0.483820047	0.635892116	0.596303502	0.536777583	0.693438914	0.716959064	0.531656548	0.734131737	0.529818496	0.505358615	0.641884817	0.576669802	0.533043478	0.681111111	0.469013007	0.554751131	0.534437663	0.465098634	0.562385321	0.494753834	0.75399754	0.707852194	0.501226492	0.493161706	0.700571429	0.837431694	0.847856155	0.770100503	0.703788749	0.5	0.601570167	0.518612521	0.716121495	0.626789366	0.749388753	0.485736926	0.544404973	0.609343936	0.816245007	0.837431694	0.586602871	0.46021021	0.586602871	0.704597701	0.837431694	0.773989899	0.745742092	0.809775429	0.468296409	0.605731225	0.554249548	0.802356021	0.650053022	0.51642797	0.608134921	0.606930693	0.635233161	0.541998232	0.820615797
C5+S4	0.550881954	0.444687842	0.442265795	0.37488458	0.383018868	0.383018868	0.536327609	0.494518879	0.434224599	0.599704579	0.62654321	0.429175476	0.646496815	0.427368421	0.403578529	0.542780749	0.474299065	0.430540827	0.585858586	0.369090909	0.452115813	0.431914894	0.365436544	0.459796149	0.393410853	0.669966997	0.616084977	0.399606299	0.391891892	0.607784431	0.773333333	0.786821705	0.689303905	0.611445783	0.398429833	0.5	0.416410256	0.625577812	0.526588846	0.664484452	0.384834123	0.441784548	0.508135169	0.746323529	0.773333333	0.484486874	0.360888889	0.484486874	0.612368024	0.773333333	0.694017094	0.660162602	0.738181818	0.368421053	0.504347826	0.451612903	0.728904847	0.551630435	0.414285714	0.506866417	0.505603985	0.535620053	0.439393939	0.751851852
C5+S5	0.632222222	0.528810409	0.526364477	0.456661316	0.465249387	0.465249387	0.618478261	0.578252033	0.518214936	0.677380952	0.701602959	0.513074842	0.719342604	0.511230907	0.486740804	0.624588364	0.558390579	0.514466546	0.664719626	0.450514648	0.536286522	0.51586582	0.446624804	0.543977055	0.476150628	0.739921977	0.692214112	0.482612383	0.474562135	0.684717208	0.827034884	0.837997054	0.756648936	0.688029021	0.481387479	0.583589744	0.5	0.700738916	0.609207709	0.735142119	0.467159278	0.525878004	0.591476091	0.804809052	0.827034884	0.568431568	0.441770186	0.568431568	0.688861985	0.827034884	0.760695187	0.731362468	0.798036466	0.449802372	0.587809917	0.535781544	0.790277778	0.632925473	0.497812773	0.590248963	0.589026915	0.617806732	0.523459062	0.809388336
C5+S6	0.423344948	0.324	0.321854305	0.264130435	0.27090301	0.27090301	0.409090909	0.369300912	0.314766839	0.472762646	0.501030928	0.310344828	0.522580645	0.308767471	0.288256228	0.415384615	0.350649351	0.311538462	0.458490566	0.259338314	0.330612245	0.312741313	0.256329114	0.3375	0.279631761	0.548532731	0.489919355	0.284876905	0.278350515	0.481188119	0.671270718	0.688385269	0.570422535	0.48502994	0.283878505	0.374422188	0.299261084	0.5	0.399671053	0.542410714	0.272421525	0.321428571	0.382075472	0.637795276	0.671270718	0.36	0.252598753	0.36	0.486	0.671270718	0.575829384	0.537610619	0.627906977	0.258785942	0.378504673	0.330163043	0.616751269	0.42408377	0.297429621	0.380877743	0.3796875	0.408403361	0.319316689	0.644562334
C5+S7	0.524425287	0.418577982	0.416191562	0.350287908	0.358194308	0.358194308	0.509776536	0.467948718	0.408277405	0.573899371	0.601317957	0.403314917	0.621805792	0.401540154	0.378238342	0.516265912	0.447852761	0.404656319	0.559815951	0.344664778	0.425904317	0.406006674	0.341121495	0.433491686	0.368314834	0.646017699	0.590614887	0.374358974	0.366834171	0.582137161	0.754132231	0.768421053	0.666058394	0.585874799	0.373210634	0.473411154	0.390792291	0.600328947	0.5	0.640350877	0.359960552	0.41571754	0.481530343	0.725646123	0.754132231	0.457967378	0.336715867	0.457967378	0.58681672	0.754132231	0.670955882	0.635888502	0.717092338	0.34401508	0.477748691	0.425407925	0.707364341	0.525179856	0.388711395	0.480263158	0.479002625	0.509065551	0.413363533	0.731462926
C5+S8	0.382462687	0.287921348	0.285913529	0.232426304	0.238649593	0.238649593	0.368705036	0.330645161	0.279291553	0.430672269	0.458612975	0.275167785	0.480093677	0.273698264	0.254658385	0.374771481	0.312977099	0.276280323	0.416666667	0.228031146	0.294117647	0.277401894	0.225274725	0.30058651	0.246690734	0.50617284	0.447598253	0.251533742	0.245508982	0.438972163	0.632716049	0.650793651	0.528350515	0.442764579	0.250611247	0.335515548	0.264857881	0.457589286	0.359649123	0.5	0.240046838	0.28551532	0.342809365	0.597667638	0.632716049	0.321821036	0.221861472	0.321821036	0.443722944	0.632716049	0.533854167	0.495169082	0.58739255	0.227524972	0.339403974	0.293696275	0.575842697	0.38317757	0.263157895	0.341666667	0.340531561	0.368043088	0.283540802	0.604719764
C6+S1	0.662244898	0.561418685	0.559000861	0.489441931	0.498081351	0.498081351	0.649	0.609962406	0.550933786	0.705434783	0.728395062	0.545836838	0.745120551	0.544006706	0.519615693	0.654894046	0.590536852	0.547217538	0.693376068	0.483246463	0.568799299	0.548605241	0.479320532	0.576376554	0.509019608	0.76442874	0.719512195	0.515488483	0.507427678	0.712403952	0.845052083	0.855072464	0.780048077	0.715545755	0.514263074	0.615165877	0.532840722	0.727578475	0.640039448	0.759953162	0.5	0.558519793	0.622840691	0.824650572	0.845052083	0.600370028	0.474415205	0.600370028	0.716335541	0.845052083	0.783816425	0.756410256	0.818411097	0.482527881	0.619274809	0.568301226	0.81125	0.662921348	0.530662306	0.62164751	0.620458891	0.648351648	0.556126821	0.828863346
C6+S2	0.607819905	0.502941176	0.500487805	0.431092437	0.439588689	0.439588689	0.59375	0.552801724	0.492322457	0.654336735	0.679470199	0.487179487	0.697959184	0.485335856	0.460916442	0.6	0.53271028	0.488571429	0.64125	0.425020713	0.510447761	0.489971347	0.421182266	0.518181818	0.450395083	0.719495091	0.669712794	0.45681211	0.448818898	0.661935484	0.811708861	0.823434992	0.737068966	0.66536965	0.455595027	0.558215452	0.474121996	0.678571429	0.58428246	0.71448468	0.441480207	0.5	0.566225166	0.788018433	0.811708861	0.542857143	0.416396104	0.542857143	0.666233766	0.811708861	0.74132948	0.710526316	0.780821918	0.424317618	0.5625	0.509940358	0.772590361	0.608540925	0.471941122	0.564977974	0.563736264	0.593063584	0.49757517	0.792890263
C6+S3	0.54281768	0.436666667	0.434254144	0.36728972	0.375358166	0.375358166	0.528225806	0.486386139	0.426247289	0.59186747	0.618897638	0.421221865	0.63902439	0.419423693	0.395770393	0.534693878	0.466192171	0.422580645	0.577941176	0.361545538	0.444067797	0.42394822	0.357923497	0.451724138	0.385672228	0.662731872	0.608359133	0.391824526	0.384164223	0.6	0.767578125	0.781312127	0.682291667	0.603686636	0.390656064	0.491864831	0.408523909	0.617924528	0.518469657	0.657190635	0.377159309	0.433774834	0.5	0.740112994	0.767578125	0.476363636	0.353417266	0.476363636	0.604615385	0.767578125	0.687062937	0.65282392	0.731843575	0.360881543	0.496212121	0.443566591	0.722426471	0.543568465	0.406411582	0.498730964	0.497468354	0.527516779	0.431394072	0.74573055
C6+S4	0.29424307	0.213953488	0.212307692	0.169325153	0.174242424	0.174242424	0.282208589	0.24954792	0.206896552	0.337408313	0.363157895	0.203539823	0.383333333	0.202346041	0.18699187	0.2875	0.234693878	0.204444444	0.324705882	0.165865385	0.219047619	0.205357143	0.163701068	0.224390244	0.180628272	0.408284024	0.352941176	0.184491979	0.1796875	0.345	0.536964981	0.556451613	0.429906542	0.348484848	0.183754993	0.253676471	0.195190948	0.362204724	0.274353877	0.402332362	0.175349428	0.211981567	0.259887006	0.5	0.536964981	0.242105263	0.161026838	0.242105263	0.349367089	0.536964981	0.43533123	0.397694524	0.489361702	0.165467626	0.25698324	0.218700475	0.477508651	0.294871795	0.193820225	0.25891182	0.257943925	0.281632653	0.210365854	0.507352941
C6+S5	0.264444444	0.190095847	0.18858954	0.149497487	0.153945666	0.153945666	0.253191489	0.222846442	0.183641975	0.305128205	0.329639889	0.180576631	0.348973607	0.179487179	0.16550765	0.25813449	0.20913884	0.181402439	0.293103448	0.146371464	0.194762684	0.182235835	0.144417476	0.19966443	0.159731544	0.373040752	0.319892473	0.163237311	0.158878505	0.312335958	0.5	0.519650655	0.394039735	0.315649867	0.162568306	0.226666667	0.172965116	0.328729282	0.245867769	0.367283951	0.154947917	0.188291139	0.232421875	0.463035019	0.5	0.215970962	0.142004773	0.215970962	0.316489362	0.5	0.399328859	0.362804878	0.452471483	0.14601227	0.22972973	0.194444444	0.440740741	0.265033408	0.171717172	0.23151751	0.230620155	0.252653928	0.186813187	0.470355731
C6+S6	0.566186107	0.460063898	0.457627119	0.389540126	0.397790055	0.397790055	0.551724138	0.510035419	0.449531738	0.614509246	0.640949555	0.444444444	0.660550459	0.442622951	0.418604651	0.558139535	0.489795918	0.445820433	0.600834492	0.38365897	0.467532468	0.447204969	0.37994723	0.475247525	0.40831758	0.683544304	0.630656934	0.414587332	0.406779661	0.622478386	0.784029038	0.79704797	0.702439024	0.626086957	0.413397129	0.515513126	0.431568432	0.64	0.542032622	0.678178964	0.399629972	0.457142857	0.523636364	0.757894737	0.784029038	0.5	0.375325804	0.5	0.626995646	0.784029038	0.707037643	0.673946958	0.75	0.382978723	0.519855596	0.467027027	0.740994854	0.566929134	0.429423459	0.522370012	0.521109771	0.551020408	0.454736842	0.763250883
C6+S7	0.684761905	0.586460033	0.584077985	0.51504298	0.523670794	0.523670794	0.671962617	0.634038801	0.576121795	0.726262626	0.74817898	0.571088165	0.764080765	0.569279493	0.545109932	0.677662582	0.615055603	0.572452229	0.71471173	0.508846426	0.593724195	0.573822825	0.50491573	0.601170569	0.534572491	0.782372144	0.739711934	0.541008277	0.532987398	0.732925586	0.857995227	0.867310012	0.797117517	0.735926305	0.53978979	0.639111111	0.558229814	0.747401247	0.663284133	0.778138528	0.525584795	0.583603896	0.646582734	0.838973162	0.857995227	0.624674196	0.5	0.624674196	0.736680328	0.857995227	0.800668151	0.774784483	0.833140209	0.508127208	0.643112701	0.593234323	0.826436782	0.685414681	0.556071152	0.645421903	0.644265233	0.671335201	0.581244947	0.842907386
C6+S8	0.566186107	0.460063898	0.457627119	0.389540126	0.397790055	0.397790055	0.551724138	0.510035419	0.449531738	0.614509246	0.640949555	0.444444444	0.660550459	0.442622951	0.418604651	0.558139535	0.489795918	0.445820433	0.600834492	0.38365897	0.467532468	0.447204969	0.37994723	0.475247525	0.40831758	0.683544304	0.630656934	0.414587332	0.406779661	0.622478386	0.784029038	0.79704797	0.702439024	0.626086957	0.413397129	0.515513126	0.431568432	0.64	0.542032622	0.678178964	0.399629972	0.457142857	0.523636364	0.757894737	0.784029038	0.5	0.375325804	0.5	0.626995646	0.784029038	0.707037643	0.673946958	0.75	0.382978723	0.519855596	0.467027027	0.740994854	0.566929134	0.429423459	0.522370012	0.521109771	0.551020408	0.454736842	0.763250883
C7+S1	0.43707483	0.336387435	0.33420026	0.2751606	0.282107574	0.282107574	0.422697368	0.382440476	0.32697201	0.486742424	0.51503006	0.322459222	0.536534447	0.320848939	0.299883314	0.429048414	0.363507779	0.323677582	0.472426471	0.270241851	0.343124166	0.324905183	0.267151767	0.35013624	0.291053228	0.562363239	0.503921569	0.296424452	0.289740699	0.495183044	0.683510638	0.70027248	0.584090909	0.499029126	0.295402299	0.387631976	0.311138015	0.514	0.41318328	0.556277056	0.283664459	0.333766234	0.395384615	0.650632911	0.683510638	0.373004354	0.263319672	0.373004354	0.5	0.683510638	0.589449541	0.551502146	0.640897756	0.269674711	0.391768293	0.342666667	0.629901961	0.437819421	0.309265945	0.394171779	0.392966361	0.422003284	0.331612903	0.657289003
C7+S2	0.264444444	0.190095847	0.18858954	0.149497487	0.153945666	0.153945666	0.253191489	0.222846442	0.183641975	0.305128205	0.329639889	0.180576631	0.348973607	0.179487179	0.16550765	0.25813449	0.20913884	0.181402439	0.293103448	0.146371464	0.194762684	0.182235835	0.144417476	0.19966443	0.159731544	0.373040752	0.319892473	0.163237311	0.158878505	0.312335958	0.5	0.519650655	0.394039735	0.315649867	0.162568306	0.226666667	0.172965116	0.328729282	0.245867769	0.367283951	0.154947917	0.188291139	0.232421875	0.463035019	0.5	0.215970962	0.142004773	0.215970962	0.316489362	0.5	0.399328859	0.362804878	0.452471483	0.14601227	0.22972973	0.194444444	0.440740741	0.265033408	0.171717172	0.23151751	0.230620155	0.252653928	0.186813187	0.470355731
C7+S3	0.350980392	0.260932945	0.259044863	0.20911215	0.214885954	0.214885954	0.337735849	0.301346801	0.252824859	0.397777778	0.425178147	0.248956885	0.44638404	0.24757953	0.229781772	0.343570058	0.284578696	0.25	0.384120172	0.205040092	0.266766021	0.251051893	0.202488688	0.272865854	0.222360248	0.472295515	0.414351852	0.226869455	0.221260816	0.405895692	0.600671141	0.619377163	0.494475138	0.409610984	0.226010101	0.305982906	0.239304813	0.424170616	0.329044118	0.466145833	0.216183575	0.25867052	0.312937063	0.56466877	0.600671141	0.292962357	0.199331849	0.292962357	0.410550459	0.600671141	0.5	0.461340206	0.554179567	0.204571429	0.309688581	0.266369048	0.542424242	0.351669941	0.237715803	0.31184669	0.310763889	0.337099812	0.256814921	0.571884984
C7+S4	0.387037037	0.291899441	0.289875173	0.235891648	0.242178447	0.242178447	0.373214286	0.334935897	0.283197832	0.435416667	0.463414634	0.279038718	0.484918794	0.277556441	0.258343634	0.379310345	0.317147193	0.280160858	0.421370968	0.23145072	0.298145506	0.281292059	0.228665208	0.304664723	0.250299401	0.511002445	0.452380952	0.255189255	0.249106079	0.44373673	0.637195122	0.655172414	0.533163265	0.447537473	0.254257908	0.339837398	0.268637532	0.462389381	0.364111498	0.504830918	0.243589744	0.289473684	0.34717608	0.602305476	0.637195122	0.326053042	0.225215517	0.326053042	0.448497854	0.637195122	0.538659794	0.5	0.592067989	0.230939227	0.34375	0.297720798	0.580555556	0.387755102	0.266922095	0.34602649	0.344884488	0.37254902	0.287482806	0.609329446
C7+S5	0.303157895	0.221198157	0.219512195	0.175395859	0.180451128	0.180451128	0.290909091	0.257602862	0.213967311	0.346987952	0.373056995	0.210526316	0.393442623	0.209302326	0.193548387	0.296296296	0.242424242	0.211453744	0.334106729	0.171837709	0.226415094	0.212389381	0.169611307	0.231884058	0.187012987	0.418604651	0.362720403	0.190981432	0.186046512	0.354679803	0.547528517	0.566929134	0.440366972	0.358208955	0.190224571	0.261818182	0.201963534	0.372093023	0.282907662	0.41260745	0.181588903	0.219178082	0.268156425	0.510638298	0.547528517	0.25	0.166859791	0.25	0.359102244	0.547528517	0.445820433	0.407932011	0.5	0.171428571	0.26519337	0.226059655	0.488135593	0.303797468	0.200557103	0.26716141	0.266173752	0.290322581	0.217522659	0.517985612
C7+S6	0.677702045	0.578553616	0.57615894	0.506919155	0.515555556	0.515555556	0.664756447	0.626462646	0.568163265	0.71975181	0.742004264	0.563106796	0.758169935	0.561290323	0.537037037	0.670520231	0.607329843	0.564476886	0.708036623	0.500719424	0.585858586	0.565853659	0.496788009	0.593350384	0.526475038	0.776785714	0.733403583	0.532924962	0.524886878	0.72651357	0.85398773	0.863523573	0.791808874	0.729559748	0.531703591	0.631578947	0.550197628	0.741214058	0.65598492	0.772475028	0.517472119	0.575682382	0.639118457	0.834532374	0.85398773	0.617021277	0.491872792	0.617021277	0.730325289	0.85398773	0.795428571	0.769060773	0.828571429	0.5	0.635616438	0.585365854	0.821723731	0.678362573	0.548031496	0.637946838	0.636779506	0.664122137	0.573311367	0.838554217
C7+S7	0.546575342	0.440397351	0.437980241	0.370817844	0.378917379	0.378917379	0.532	0.49017199	0.429956897	0.595522388	0.622464899	0.424920128	0.642512077	0.423117709	0.399399399	0.538461538	0.469964664	0.426282051	0.581632653	0.36505032	0.447811448	0.427652733	0.361413043	0.455479452	0.389268293	0.666110184	0.61196319	0.395441031	0.387755102	0.603630862	0.77027027	0.78388998	0.68556701	0.607305936	0.394268775	0.495652174	0.412190083	0.621495327	0.522251309	0.660596026	0.380725191	0.4375	0.503787879	0.74301676	0.77027027	0.480144404	0.356887299	0.480144404	0.608231707	0.77027027	0.690311419	0.65625	0.73480663	0.364383562	0.5	0.447309417	0.725454545	0.547325103	0.410071942	0.502518892	0.501256281	0.531291611	0.435114504	0.748592871
C7+S8	0.598300971	0.493	0.490547264	0.421367521	0.429816914	0.429816914	0.584123223	0.542951542	0.482387476	0.645287958	0.670748299	0.477250726	0.68951049	0.475409836	0.45105215	0.590419162	0.522799576	0.478640777	0.632051282	0.415332772	0.500507614	0.480038948	0.411519199	0.508247423	0.440571939	0.711399711	0.660857909	0.446962829	0.439002671	0.652980132	0.805555556	0.817578773	0.729289941	0.656458056	0.445750452	0.548387097	0.464218456	0.669836957	0.574592075	0.706303725	0.431698774	0.490059642	0.556433409	0.781299525	0.805555556	0.532972973	0.406765677	0.532972973	0.657333333	0.805555556	0.733630952	0.702279202	0.773940345	0.414634146	0.552690583	0.5	0.76552795	0.599027947	0.462043112	0.55518018	0.553932584	0.583431953	0.487636004	0.786283892
C8+S1	0.313278008	0.229483283	0.22775264	0.18236715	0.18757764	0.18757764	0.300796813	0.266784452	0.222058824	0.357819905	0.384223919	0.218523878	0.404825737	0.217266187	0.201065246	0.306288032	0.25124792	0.219476744	0.344748858	0.178698225	0.234836703	0.220437956	0.176401869	0.24044586	0.194337194	0.43019943	0.373762376	0.198423127	0.193341869	0.365617433	0.559259259	0.578544061	0.452095808	0.369193154	0.197643979	0.271095153	0.209722222	0.383248731	0.292635659	0.424157303	0.18875	0.227409639	0.277573529	0.522491349	0.559259259	0.259005146	0.173563218	0.259005146	0.370098039	0.559259259	0.457575758	0.419444444	0.511864407	0.178276269	0.274545455	0.23447205	0.5	0.313929314	0.208275862	0.276556777	0.275547445	0.300198807	0.225710015	0.529824561
C8+S2	0.49924357	0.394265233	0.39192399	0.327706058	0.335365854	0.335365854	0.484581498	0.44295302	0.384167637	0.549084859	0.576923077	0.379310345	0.597826087	0.377574371	0.35483871	0.491071429	0.423076923	0.380622837	0.534846029	0.322265625	0.401459854	0.381944444	0.31884058	0.408921933	0.345188285	0.622641509	0.566037736	0.35106383	0.34375	0.557432432	0.734966592	0.75	0.643274854	0.56122449	0.349946978	0.448369565	0.367074527	0.57591623	0.474820144	0.61682243	0.337078652	0.391459075	0.456431535	0.705128205	0.734966592	0.433070866	0.314585319	0.433070866	0.562180579	0.734966592	0.648330059	0.612244898	0.696202532	0.321637427	0.452674897	0.400972053	0.686070686	0.5	0.365044248	0.455172414	0.45392022	0.483870968	0.389150943	0.711206897
C8+S3	0.634254144	0.530989824	0.52854512	0.458832934	0.46742671	0.46742671	0.620540541	0.580384226	0.520398912	0.679289941	0.703431373	0.515260323	0.721105528	0.513416816	0.488926746	0.626637555	0.560546875	0.516651665	0.666666667	0.452681388	0.538461538	0.518050542	0.448788116	0.546146527	0.478333333	0.741602067	0.69407497	0.484797297	0.476744186	0.686602871	0.828282828	0.839181287	0.758256275	0.689903846	0.48357203	0.585714286	0.502187227	0.702570379	0.611288605	0.736842105	0.469337694	0.528058878	0.593588418	0.806179775	0.828282828	0.570576541	0.443928848	0.570576541	0.690734055	0.828282828	0.762284197	0.733077905	0.799442897	0.451968504	0.589928058	0.537956888	0.791724138	0.634955752	0.5	0.592363261	0.591143151	0.61987041	0.525641026	0.810734463
C8+S4	0.544077135	0.437915743	0.435501654	0.368470149	0.376549094	0.376549094	0.529490617	0.487654321	0.427489177	0.593093093	0.620094192	0.422459893	0.640194489	0.420660277	0.396984925	0.535956581	0.467455621	0.423819742	0.579178886	0.36271809	0.445321308	0.425188375	0.359090909	0.452981651	0.386875612	0.663865546	0.609567901	0.393034826	0.385365854	0.601217656	0.76848249	0.782178218	0.683391003	0.604900459	0.391865079	0.493133583	0.409751037	0.619122257	0.519736842	0.658333333	0.37835249	0.435022026	0.501269036	0.74108818	0.76848249	0.477629988	0.354578097	0.477629988	0.605828221	0.76848249	0.68815331	0.65397351	0.73283859	0.362053162	0.497481108	0.44481982	0.723443223	0.544827586	0.407636739	0.5	0.498737374	0.528781794	0.43263965	0.746691871
C8+S5	0.54532967	0.439159292	0.436743674	0.369646182	0.37773549	0.37773549	0.530748663	0.488916256	0.428725702	0.594311377	0.621283255	0.423692636	0.641357027	0.421891605	0.398194584	0.537212449	0.468713105	0.425053533	0.580409357	0.363886343	0.446569179	0.426423201	0.360254083	0.45423341	0.388074291	0.664991625	0.610769231	0.394240318	0.386562804	0.602427921	0.769379845	0.783037475	0.684482759	0.60610687	0.393069307	0.494396015	0.410973085	0.6203125	0.520997375	0.659468439	0.379541109	0.436263736	0.502531646	0.742056075	0.769379845	0.478890229	0.355734767	0.478890229	0.607033639	0.769379845	0.689236111	0.655115512	0.733826248	0.363220494	0.498743719	0.446067416	0.724452555	0.54607978	0.408856849	0.501262626	0.5	0.530040053	0.433879781	0.747645951
C8+S6	0.515373353	0.409778813	0.407407407	0.342079689	0.349900596	0.349900596	0.500711238	0.4589309	0.39954597	0.565008026	0.592592593	0.394618834	0.613240418	0.392857143	0.369747899	0.507204611	0.438902743	0.395950506	0.55086072	0.336520076	0.417061611	0.397291196	0.333017975	0.424607961	0.3599182	0.637681159	0.581818182	0.365904366	0.358452138	0.573289902	0.747346072	0.761904762	0.657943925	0.57704918	0.364766839	0.464379947	0.382193268	0.591596639	0.490934449	0.631956912	0.351648352	0.406936416	0.472483221	0.718367347	0.747346072	0.448979592	0.328664799	0.448979592	0.577996716	0.747346072	0.662900188	0.62745098	0.709677419	0.335877863	0.468708389	0.416568047	0.699801193	0.516129032	0.38012959	0.471218206	0.469959947	0.5	0.404597701	0.724279835
C8+S7	0.610129564	0.505365854	0.502912621	0.433472803	0.441979522	0.441979522	0.596087457	0.555198285	0.494746896	0.65652725	0.681578947	0.489603025	0.7	0.487758945	0.46332737	0.602325581	0.535123967	0.490995261	0.643478261	0.427392739	0.512871287	0.492395437	0.423548651	0.520603015	0.452797203	0.721448468	0.671854734	0.459219858	0.451219512	0.664102564	0.813186813	0.824840764	0.738944365	0.667525773	0.458001768	0.560606061	0.476540938	0.680683311	0.586636467	0.716459198	0.443873179	0.50242483	0.568605928	0.789634146	0.813186813	0.545263158	0.418755053	0.545263158	0.668387097	0.813186813	0.743185079	0.712517194	0.782477341	0.426688633	0.564885496	0.512363996	0.774289985	0.610849057	0.474358974	0.56736035	0.566120219	0.595402299	0.5	0.794478528
C8+S8	0.288172043	0.209048362	0.207430341	0.165228113	0.170050761	0.170050761	0.27628866	0.244080146	0.202111614	0.330864198	0.356382979	0.198813056	0.376404494	0.197640118	0.182561308	0.281512605	0.229452055	0.199701937	0.318289786	0.161835749	0.214057508	0.200598802	0.159713945	0.219312602	0.176315789	0.401197605	0.34625323	0.180107527	0.17539267	0.338383838	0.529644269	0.549180328	0.422712934	0.341836735	0.179384203	0.248148148	0.190611664	0.355437666	0.268537074	0.395280236	0.171136654	0.207109737	0.25426945	0.492647059	0.529644269	0.236749117	0.157092614	0.236749117	0.342710997	0.529644269	0.428115016	0.390670554	0.482014388	0.161445783	0.251407129	0.213716108	0.470175439	0.288793103	0.189265537	0.253308129	0.252354049	0.275720165	0.205521472	0.5

You’ll notice that starting with Row 1, Column 1 (position [1,1]) and following along a perfect diagonal (positions [2,2], [3,3], [4,4], etc.) all the probabilities are 0.5, or dead even. This is because due to the way the table is set up, each C_n+S_n is pitted against itself exactly once.

There are also 448 other instances of the same Competitor facing itself with a different Strategy.

These 512 matchups notwithstanding, our table does give us the win probabilities for every real combination of C_n+S_n.

This is Bayes’ Theorem at work.

We’re on to a start.