[tool21]

I always like to include examples of what i'm talking about and this one is best described as one. I just want the BEST math behind this and the right way to do it. I have tried about 4 different methods and i have my favorites but i want to know how you guys would do it. This is just something small that i need to zero in on for accuracy.



Answer this question and how do you explain how to do it.

If Team A's offense averages 78.2 points against teams that give up on average 71.96667 and Team B's defense gives up 66.1 points against teams that put up on average 75.0875 then what is a good estimation about how many points Team A will put up based on the given information?

Team A
78.2
71.9667

Team B
66.1
75.0875

The problem i'm coming up with is with the denominator. If team A difference is +6.233 and Team B difference is -8.98 then i come up with -2.75 against both teams opponents but which denominator (opp) do i add this to? Do i take the average of the denominators and then add it to get 70.78 points for Team A?

Another method is this: Team A averages 78.2 against teams that give up on average 71.9667 but Team B doesn't give up 71.9667 they give up 66.1 so Team A off will have 71.825. On the other side, Team B gives up on average 66.1 against teams that average 75.0875 but Team A doesn't put up 75.0875 they put up 78.2 so Team B def. will give up 68.84. Then take the average of 68.84 and 71.825.

Obviously there IS a distinct difference in the difficulty of the opponents and there is a difference between 68.84 points and 71.825 points in the 2nd example. Should i just stick to games in which the opponents difficulty is fairly the same? Divisonal games in NCAAB, and NBA mid-season? NBA isn't that much of a problem because the teams play all of each other. The denominators are fairly close which leads to less error but IMO NBA lines are a lot sharper than college lines. College is where the problem lies. Take a team like Davidson playing Kansas. Slight variations in opponents difficulty.

Another problem between these 2 ways that i found is when two teams both like to run. Say you have Phoenix playing Denver. If you use the 2nd example method you are taking the averages which leads to a false total. Say team A is +6 and team B defense is +4 then you would have +10 to the teams they've played against. In example 2 the average would be taken between the two numbers and a false total would come out. So instead of having a 117-112 score (method 2) you should really have a 124-119 (method 1) type score. But like i asked before, which denominator do I add the numbers to? So I more or less just need help with the first example because the 2nd gives false totals....sorry for all the info just thinking on paper.


Just throwing out some ideas, i have various others that tweak this....what would you guys do?

[BuddyBear]

I didn't read the entire message b/c it's too long, but based on the first paragraph and the title of this thread you are interested in controlling for the quality of competition.

I do not think the way you are considering this issue by looking at point differential is very valid but some might argue that point.

Well, basically, you would need to develop a variable that measures for opponent's strength of schedule and treat that as a stand alone primary variable in your model or you could simply treat it as a control variable...it would be up to you how you wanted to use it but I certainly agree that it is important.

IMO, this is an extremely difficult variable to construct because there are several validity issues that need to be considered before doing this. The biggest obstacle with this variable is that it is ALWAYS varying and the earlier in a season the more volatile it is going to be. Really, it won't become stagnant until you get deep into the season and everyone has played everyone.

There are a number of reputable websites that offer strength of schedule measures that you can consider. It will save you a ton of time.

[Ganchrow]

BuddyBear brings up some very good points.

He's particularly correct that without prior knowledge of the sport in question (in other words, additional data and/or theory), point estimates of scoring expectations are by themselves generally insufficient to provide the answer for which you're looking.

Off-the-cuff I'd say that to really go about this theoretically correctly ("BEST math", as you say) you'd first want to model the pdf of each team's scoring (for or against as applicable) conditioned on some measure of its opponent's performance. You'd further want to model a marginal (i.e., a "prior") probability distribution for each teams' (offensive or defensive) performance (assuming a one-to-one relationship existed between a team's marginal performance and it's scoring). Finally you'd utilize Bayesian inference to combine all your estimates in order to determine a posterior pdf for scoring by team A conditioned on the joint realization of your predetermined performance measures. By integrating across all possible performance measure realizations you'd then be able to calculate a final expectation. This would be what I'd consider a serious project, with likely considerably overreaching complexity.

If, however, you'd settle for roughly estimating Team A's win probability vs. Team B, then there is a straightforward (if slightly involved) methodology that may be employed to this end. Still necessary, however, is the assumption that the opponents of Team A and B have themselves played against directly relatable opposition (consider that there'd be no way one would be able to draw meaningful conclusions about the Knicks' likely performance against a Division III team based solely on the two teams' results within their respective leagues).

I'll warn you, however, that the results at best represent a first-order approximation far better suited to Rotisserie League play than to professional advantage betting. Nevertheless, the general techniques involved may be instructive and could potentially prove useful within alternate contexts.

Given by OP:

Team A's points for = 78.2
Team A's opps' points against = 71.96667
Team B's points against = 66.1
Team B's opps' points for = 75.0875

Let's add a few more variables to the problem specification, namely:

Team A's points against = w
Team A's opps' points for = x
Team B's points for = y
Team B's opps' points against = z

Baseball's Bill James posited that the number of wins expected by a baseball team over some stretch of games could be reasonably estimated as a function solely of total runs scored and total runs allowed. This is known as the "Pythagorean expectation" and is given by the following formula:

Expected MLB Wins = Games Played * RunsFor^n/(RunsFor^n + RunsAgainst^n)

where n is a data-determined exponent equal to roughly 2 for Major League Baseball.

It's certainly possible (although not necessarily advisable) to apply this same methodology to other sports. I'm going to demonstrate an extremely simple exponent estimation technique for the NBA. Disinterested readers won't lose much by skipping ahead to the conclusion.

Using covers.com data one can derive the following data set through the 2006-07 season (also see Sheet1 of the attached spreadsheet):

Team	Points For	Points Against	Wins	Games
ATLANTA1991	9,489	9,466	45	87
ATLANTA1992	8,711	8,834	38	82
ATLANTA1993	9,101	9,214	43	85
ATLANTA1994	9,321	8,855	63	93
ATLANTA1995	8,189	8,116	42	85
ATLANTA1996	8,315	8,262	46	85
ATLANTA1997	8,204	7,813	56	87
ATLANTA1998	8,210	7,916	51	86
ATLANTA1999	5,032	4,912	34	59
ATLANTA2000	7,735	8,176	28	82
ATLANTA2001	7,459	7,886	25	82
ATLANTA2002	7,711	8,058	33	82
ATLANTA2003	7,714	8,006	35	82
ATLANTA2004	7,611	7,992	28	82
ATLANTA2005	7,605	8,401	13	82
ATLANTA2006	7,972	8,362	26	82
ATLANTA2007	7,680	8,070	30	82
BOSTON1991	10,359	9,869	61	93
BOSTON1992	9,816	9,518	57	92
BOSTON1993	8,904	8,852	49	86
BOSTON1994	8,267	8,618	32	82
BOSTON1995	8,773	8,975	36	86
BOSTON1996	8,002	8,258	30	77
BOSTON1997	7,941	8,567	13	79
BOSTON1998	7,864	8,079	36	82
BOSTON1999	4,650	4,743	19	50
BOSTON2000	8,146	8,208	35	82
BOSTON2001	7,759	7,934	36	82
BOSTON2002	9,361	9,135	58	98
BOSTON2003	8,545	8,583	48	92
BOSTON2004	8,149	8,335	36	86
BOSTON2005	8,918	8,851	48	89
BOSTON2006	8,033	8,159	33	82
BOSTON2007	7,857	8,137	24	82
CHARLOTTE2005	7,729	8,220	18	82
CHARLOTTE2006	7,943	8,270	26	82
CHARLOTTE2007	7,945	8,252	33	82
CHICAGO1991	10,791	9,847	76	99
CHICAGO1992	11,219	10,227	82	104
CHICAGO1993	10,570	9,943	72	101
CHICAGO1994	8,962	8,696	60	92
CHICAGO1995	9,306	8,916	52	92
CHICAGO1996	9,605	8,457	82	92
CHICAGO1997	9,689	8,702	80	95
CHICAGO1998	9,887	9,157	77	103
CHICAGO1999	4,095	4,568	13	50
CHICAGO2000	6,952	7,723	17	82
CHICAGO2001	7,181	7,927	15	82
CHICAGO2002	7,335	8,035	21	82
CHICAGO2003	7,786	8,207	30	82
CHICAGO2004	7,355	7,876	23	82
CHICAGO2005	8,360	8,284	49	88
CHICAGO2006	8,610	8,576	43	88
CHICAGO2007	9,024	8,602	55	92
CLEVELAND1991	8,343	8,545	33	82
CLEVELAND1992	10,687	10,197	66	99
CLEVELAND1993	9,675	9,165	57	91
CLEVELAND1994	8,580	8,270	47	85
CLEVELAND1995	7,747	7,729	44	86
CLEVELAND1996	6,718	6,598	40	74
CLEVELAND1997	6,867	6,713	40	78
CLEVELAND1998	7,908	7,716	48	86
CLEVELAND1999	4,322	4,408	22	50
CLEVELAND2000	7,950	8,237	32	82
CLEVELAND2001	7,561	7,909	30	82
CLEVELAND2002	7,812	8,085	29	82
CLEVELAND2003	7,495	8,284	17	82
CLEVELAND2004	7,619	7,834	35	82
CLEVELAND2005	7,914	7,849	42	82
CLEVELAND2006	9,177	9,035	57	95
CLEVELAND2007	9,710	9,353	62	102
DALLAS1991	8,195	8,570	28	82
DALLAS1992	8,007	8,634	22	82
DALLAS1993	8,141	9,387	11	82
DALLAS1994	7,811	8,514	13	82
DALLAS1995	8,463	8,700	36	82
DALLAS1996	5,632	5,846	19	55
DALLAS1997	6,921	7,374	23	76
DALLAS1998	7,494	7,995	20	82
DALLAS1999	4,581	4,701	19	50
DALLAS2000	8,316	8,363	40	82
DALLAS2001	9,161	8,847	57	92
DALLAS2002	9,496	9,142	61	90
DALLAS2003	10,526	9,898	70	102
DALLAS2004	9,124	8,753	53	87
DALLAS2005	9,772	9,322	64	95
DALLAS2006	10,424	9,832	74	105
DALLAS2007	8,792	8,240	69	88
DENVER1991	9,828	10,723	20	82
DENVER1992	8,176	8,821	24	82
DENVER1993	8,626	8,769	36	82
DENVER1994	9,369	9,263	48	94
DENVER1995	8,588	8,565	41	85
DENVER1996	6,639	6,841	29	68
DENVER1997	7,439	7,906	20	76
DENVER1998	7,300	8,266	11	82
DENVER1999	4,674	5,004	14	50
DENVER2000	8,115	8,289	35	82
DENVER2001	7,918	8,120	40	82
DENVER2002	7,559	8,036	27	82
DENVER2003	6,900	7,580	17	82
DENVER2004	8,425	8,356	44	87
DENVER2005	8,612	8,497	50	87
DENVER2006	8,664	8,683	45	87
DENVER2007	9,080	8,977	46	87
DETROIT1991	9,721	9,470	57	97
DETROIT1992	8,439	8,408	49	87
DETROIT1993	8,252	8,366	40	82
DETROIT1994	7,949	8,587	20	82
DETROIT1995	8,053	8,651	28	82
DETROIT1996	7,671	7,503	44	80
DETROIT1997	7,719	7,324	52	82
DETROIT1998	7,721	7,592	37	82
DETROIT1999	4,914	4,758	31	55
DETROIT2000	8,722	8,635	42	85
DETROIT2001	7,837	7,976	32	82
DETROIT2002	8,563	8,390	54	92
DETROIT2003	9,021	8,702	58	99
DETROIT2004	9,391	8,765	70	105
DETROIT2005	9,911	9,476	69	107
DETROIT2006	9,591	8,987	74	100
DETROIT2007	9,334	8,946	63	98
GOLDEN STATE1991	10,594	10,473	48	91
GOLDEN STATE1992	10,200	9,878	56	86
GOLDEN STATE1993	9,007	9,071	34	82
GOLDEN STATE1994	9,192	9,069	50	85
GOLDEN STATE1995	8,661	9,122	26	82
GOLDEN STATE1996	7,653	7,759	34	75
GOLDEN STATE1997	7,794	8,140	29	78
GOLDEN STATE1998	7,237	7,985	19	82
GOLDEN STATE1999	4,416	4,541	21	50
GOLDEN STATE2000	7,834	8,512	19	82
GOLDEN STATE2001	7,584	8,326	17	82
GOLDEN STATE2002	8,009	8,452	21	82
GOLDEN STATE2003	8,400	8,493	38	82
GOLDEN STATE2004	7,649	7,709	37	82
GOLDEN STATE2005	8,094	8,271	34	82
GOLDEN STATE2006	8,076	8,187	34	82
GOLDEN STATE2007	9,910	9,919	47	93
HOUSTON1991	9,033	8,763	52	85
HOUSTON1992	8,366	8,507	42	82
HOUSTON1993	9,704	9,339	61	94
HOUSTON1994	9,943	9,485	70	98
HOUSTON1995	10,389	10,182	58	100
HOUSTON1996	7,305	7,145	44	72
HOUSTON1997	9,068	8,711	62	90
HOUSTON1998	8,522	8,618	43	87
HOUSTON1999	5,099	4,992	32	54
HOUSTON2000	8,156	8,227	34	82
HOUSTON2001	7,972	7,784	45	82
HOUSTON2002	7,572	7,973	28	82
HOUSTON2003	7,689	7,567	43	82
HOUSTON2004	7,785	7,670	46	87
HOUSTON2005	8,479	8,167	54	89
HOUSTON2006	7,387	7,517	34	82
HOUSTON2007	8,564	8,188	55	89
INDIANA1991	9,751	9,785	43	87
INDIANA1992	9,520	9,387	40	85
INDIANA1993	9,247	9,107	42	86
INDIANA1994	9,701	9,404	56	98
INDIANA1995	9,836	9,467	63	99
INDIANA1996	7,685	7,425	49	77
INDIANA1997	7,487	7,467	36	79
INDIANA1998	9,342	8,807	68	98
INDIANA1999	5,950	5,716	42	63
INDIANA2000	10,562	10,124	69	105
INDIANA2001	7,940	7,981	42	86
INDIANA2002	8,396	8,373	44	87
INDIANA2003	8,487	8,235	50	88
INDIANA2004	8,861	8,318	71	98
INDIANA2005	8,718	8,694	50	95
INDIANA2006	8,235	8,102	43	88
INDIANA2007	7,840	8,040	35	82
L.A. CLIPPERS1991	8,491	8,774	31	82
L.A. CLIPPERS1992	8,931	8,863	47	87
L.A. CLIPPERS1993	9,220	9,232	43	87
L.A. CLIPPERS1994	8,447	8,916	27	82
L.A. CLIPPERS1995	7,937	8,678	17	82
L.A. CLIPPERS1996	7,463	7,754	26	75
L.A. CLIPPERS1997	7,866	8,150	32	81
L.A. CLIPPERS1998	7,865	8,469	17	82
L.A. CLIPPERS1999	4,519	4,960	9	50
L.A. CLIPPERS2000	7,546	8,491	15	82
L.A. CLIPPERS2001	7,581	7,818	31	82
L.A. CLIPPERS2002	7,844	7,884	39	82
L.A. CLIPPERS2003	7,693	8,032	27	82
L.A. CLIPPERS2004	7,770	8,147	28	82
L.A. CLIPPERS2005	7,849	7,912	37	82
L.A. CLIPPERS2006	9,238	9,064	54	94
L.A. CLIPPERS2007	7,843	7,881	40	82
L.A. LAKERS1991	10,691	10,117	70	101
L.A. LAKERS1992	8,607	8,756	44	86
L.A. LAKERS1993	9,041	9,154	41	87
L.A. LAKERS1994	8,233	8,585	33	82
L.A. LAKERS1995	9,521	9,591	53	92
L.A. LAKERS1996	7,796	7,526	46	76
L.A. LAKERS1997	8,024	7,760	52	81
L.A. LAKERS1998	9,955	9,304	68	95
L.A. LAKERS1999	5,702	5,574	34	58
L.A. LAKERS2000	10,562	9,807	82	105
L.A. LAKERS2001	9,905	9,424	71	98
L.A. LAKERS2002	10,161	9,507	73	101
L.A. LAKERS2003	9,433	9,239	56	94
L.A. LAKERS2004	9,991	9,651	69	104
L.A. LAKERS2005	8,095	8,338	34	82
L.A. LAKERS2006	8,858	8,700	48	89
L.A. LAKERS2007	8,964	9,022	43	87
MEMPHIS1996	6,515	7,207	13	72
MEMPHIS1997	6,667	7,474	12	75
MEMPHIS1998	7,923	8,522	19	82
MEMPHIS1999	4,443	4,876	8	50
MEMPHIS2000	7,702	8,163	22	82
MEMPHIS2001	7,522	7,992	23	82
MEMPHIS2002	7,366	7,978	23	82
MEMPHIS2003	7,995	8,260	28	82
MEMPHIS2004	8,264	8,120	50	86
MEMPHIS2005	8,072	7,928	45	86
MEMPHIS2006	7,895	7,648	49	86
MEMPHIS2007	8,331	8,753	22	82
MIAMI1991	8,349	8,840	24	82
MIAMI1992	8,906	9,305	38	85
MIAMI1993	8,495	8,599	36	82
MIAMI1994	8,924	8,739	44	87
MIAMI1995	8,293	8,428	32	82
MIAMI1996	7,051	6,954	36	73
MIAMI1997	8,773	8,314	66	94
MIAMI1998	8,224	7,831	57	87
MIAMI1999	4,844	4,616	35	55
MIAMI2000	8,571	8,291	58	92
MIAMI2001	7,524	7,403	50	85
MIAMI2002	7,150	7,274	36	82
MIAMI2003	7,016	7,430	25	82
MIAMI2004	8,495	8,450	48	95
MIAMI2005	9,797	9,198	70	97
MIAMI2006	10,405	10,001	68	105
MIAMI2007	8,114	8,233	44	86
MILWAUKEE1991	9,029	8,860	48	85
MILWAUKEE1992	8,608	8,749	31	82
MILWAUKEE1993	8,391	8,699	28	82
MILWAUKEE1994	7,949	8,480	20	82
MILWAUKEE1995	8,160	8,497	34	82
MILWAUKEE1996	7,451	7,878	24	78
MILWAUKEE1997	7,142	7,291	29	75
MILWAUKEE1998	7,748	7,905	36	82
MILWAUKEE1999	4,870	4,818	28	53
MILWAUKEE2000	8,780	8,753	44	87
MILWAUKEE2001	9,982	9,639	62	100
MILWAUKEE2002	7,996	8,014	41	82
MILWAUKEE2003	8,744	8,752	44	88
MILWAUKEE2004	8,467	8,442	42	87
MILWAUKEE2005	7,973	8,218	30	82
MILWAUKEE2006	8,508	8,641	41	87
MILWAUKEE2007	8,172	8,531	28	82
MINNESOTA1991	8,169	8,491	29	82
MINNESOTA1992	8,237	8,815	15	82
MINNESOTA1993	8,042	8,684	19	82
MINNESOTA1994	7,930	8,498	20	82
MINNESOTA1995	7,716	8,464	21	82
MINNESOTA1996	6,886	7,332	20	71
MINNESOTA1997	7,703	7,855	38	80
MINNESOTA1998	8,741	8,712	47	87
MINNESOTA1999	4,969	4,975	26	54
MINNESOTA2000	8,420	8,221	51	86
MINNESOTA2001	8,310	8,225	48	86
MINNESOTA2002	8,451	8,206	50	85
MINNESOTA2003	8,648	8,516	53	88
MINNESOTA2004	9,408	8,959	68	100
MINNESOTA2005	7,934	7,815	44	82
MINNESOTA2006	7,522	7,676	33	82
MINNESOTA2007	7,877	8,178	32	82
NEW JERSEY1991	8,441	8,811	26	82
NEW JERSEY1992	9,048	9,220	41	86
NEW JERSEY1993	8,899	8,819	45	87
NEW JERSEY1994	8,807	8,656	46	86
NEW JERSEY1995	8,042	8,299	30	82
NEW JERSEY1996	6,943	7,249	28	74
NEW JERSEY1997	7,432	7,791	24	76
NEW JERSEY1998	8,461	8,353	43	85
NEW JERSEY1999	4,569	4,758	16	50
NEW JERSEY2000	8,036	8,120	31	82
NEW JERSEY2001	7,552	7,966	26	82
NEW JERSEY2002	9,796	9,456	63	102
NEW JERSEY2003	9,693	9,198	63	102
NEW JERSEY2004	8,371	8,139	54	93
NEW JERSEY2005	7,883	8,057	42	86
NEW JERSEY2006	8,727	8,625	54	93
NEW JERSEY2007	9,083	9,124	47	94
NEW ORLEANS1991	8,428	8,858	26	82
NEW ORLEANS1992	8,980	9,300	31	82
NEW ORLEANS1993	9,930	9,955	48	91
NEW ORLEANS1994	8,732	8,750	41	82
NEW ORLEANS1995	8,619	8,366	51	86
NEW ORLEANS1996	7,517	7,595	36	73
NEW ORLEANS1997	7,634	7,496	50	77
NEW ORLEANS1998	8,668	8,558	55	91
NEW ORLEANS1999	4,644	4,649	26	50
NEW ORLEANS2000	8,437	8,229	50	86
NEW ORLEANS2001	8,496	8,261	52	92
NEW ORLEANS2002	8,565	8,486	48	91
NEW ORLEANS2003	8,256	8,093	49	88
NEW ORLEANS2004	8,092	8,122	44	89
NEW ORLEANS2005	7,252	7,832	18	82
NEW ORLEANS2006	7,611	7,842	38	82
NEW ORLEANS2007	7,833	7,962	39	82
NEW YORK1991	8,713	8,792	39	85
NEW YORK1992	9,411	9,080	57	94
NEW YORK1993	9,795	9,293	69	97
NEW YORK1994	9,695	9,084	69	100
NEW YORK1995	9,081	8,782	61	93
NEW YORK1996	7,589	7,410	46	79
NEW YORK1997	8,265	7,999	60	87
NEW YORK1998	8,395	8,215	47	92
NEW YORK1999	6,020	5,929	39	70
NEW YORK2000	8,906	8,803	59	98
NEW YORK2001	7,720	7,520	50	87
NEW YORK2002	7,514	7,839	30	82
NEW YORK2003	7,860	7,971	37	82
NEW YORK2004	7,878	8,050	39	86
NEW YORK2005	7,977	8,177	33	82
NEW YORK2006	7,842	8,367	23	82
NEW YORK2007	7,994	8,228	33	82
ORLANDO1991	8,684	9,010	31	82
ORLANDO1992	8,330	8,897	21	82
ORLANDO1993	8,653	8,543	41	82
ORLANDO1994	8,941	8,638	50	85
ORLANDO1995	10,790	10,200	67	99
ORLANDO1996	8,538	8,113	61	82
ORLANDO1997	7,577	7,658	42	81
ORLANDO1998	7,387	7,475	41	82
ORLANDO1999	4,818	4,713	34	54
ORLANDO2000	8,206	8,150	41	82
ORLANDO2001	8,403	8,345	44	86
ORLANDO2002	8,615	8,506	45	86
ORLANDO2003	8,690	8,731	45	89
ORLANDO2004	7,711	8,287	21	82
ORLANDO2005	8,160	8,344	36	82
ORLANDO2006	7,784	7,872	36	82
ORLANDO2007	8,123	8,095	40	86
PHILADELPHIA1991	9,448	9,473	48	90
PHILADELPHIA1992	8,358	8,462	35	82
PHILADELPHIA1993	8,556	9,029	26	82
PHILADELPHIA1994	8,033	8,658	25	82
PHILADELPHIA1995	7,820	8,236	24	82
PHILADELPHIA1996	7,450	8,275	16	79
PHILADELPHIA1997	8,131	8,658	22	81
PHILADELPHIA1998	7,651	7,847	31	82
PHILADELPHIA1999	5,197	5,090	31	58
PHILADELPHIA2000	8,713	8,616	54	92
PHILADELPHIA2001	9,887	9,538	68	105
PHILADELPHIA2002	7,906	7,819	45	87
PHILADELPHIA2003	9,046	8,847	54	94
PHILADELPHIA2004	7,215	7,419	33	82
PHILADELPHIA2005	8,582	8,683	44	87
PHILADELPHIA2006	8,147	8,307	38	82
PHILADELPHIA2007	7,785	8,033	35	82
PHOENIX1991	9,731	9,240	56	86
PHOENIX1992	10,142	9,644	57	90
PHOENIX1993	11,813	11,257	75	106
PHOENIX1994	9,923	9,578	61	92
PHOENIX1995	10,183	9,824	65	92
PHOENIX1996	7,681	7,679	38	74
PHOENIX1997	8,623	8,660	39	84
PHOENIX1998	8,538	8,143	57	86
PHOENIX1999	5,056	4,974	27	53
PHOENIX2000	8,897	8,502	57	91
PHOENIX2001	8,064	7,920	52	86
PHOENIX2002	7,802	7,856	36	82
PHOENIX2003	8,345	8,283	46	88
PHOENIX2004	7,723	8,029	29	82
PHOENIX2005	10,734	10,087	71	97
PHOENIX2006	11,030	10,551	64	102
PHOENIX2007	10,182	9,528	67	93
PORTLAND1991	11,069	10,336	72	98
PORTLAND1992	11,444	10,778	70	103
PORTLAND1993	9,297	9,033	52	86
PORTLAND1994	9,210	9,015	48	86
PORTLAND1995	8,752	8,491	44	85
PORTLAND1996	7,102	7,051	36	72
PORTLAND1997	7,909	7,573	49	80
PORTLAND1998	8,133	8,035	47	86
PORTLAND1999	5,862	5,550	42	63
PORTLAND2000	9,470	8,870	69	98
PORTLAND2001	8,091	7,791	50	85
PORTLAND2002	8,191	7,965	49	85
PORTLAND2003	8,511	8,290	53	89
PORTLAND2004	7,439	7,544	41	82
PORTLAND2005	7,621	7,949	27	82
PORTLAND2006	7,285	8,060	21	82
PORTLAND2007	7,717	8,069	32	82
SACRAMENTO1991	7,928	8,484	25	82
SACRAMENTO1992	8,549	9,046	29	82
SACRAMENTO1993	8,847	9,103	25	82
SACRAMENTO1994	8,291	8,764	28	82
SACRAMENTO1995	8,060	8,134	39	82
SACRAMENTO1996	7,031	7,264	31	71
SACRAMENTO1997	7,238	7,508	30	75
SACRAMENTO1998	7,641	8,101	27	82
SACRAMENTO1999	5,462	5,507	29	55
SACRAMENTO2000	9,089	8,890	46	87
SACRAMENTO2001	9,123	8,646	58	90
SACRAMENTO2002	10,193	9,535	71	98
SACRAMENTO2003	9,635	9,076	66	94
SACRAMENTO2004	9,575	9,163	62	94
SACRAMENTO2005	9,016	8,861	51	87
SACRAMENTO2006	8,690	8,621	46	88
SACRAMENTO2007	8,303	8,451	33	82
SAN ANTONIO1991	9,213	8,863	56	86
SAN ANTONIO1992	8,834	8,589	47	85
SAN ANTONIO1993	9,659	9,439	54	92
SAN ANTONIO1994	8,554	8,156	56	86
SAN ANTONIO1995	10,219	9,659	71	97
SAN ANTONIO1996	8,473	8,037	56	83
SAN ANTONIO1997	6,182	6,691	17	67
SAN ANTONIO1998	8,413	8,057	60	91
SAN ANTONIO1999	6,143	5,617	52	67
SAN ANTONIO2000	8,213	7,731	54	86
SAN ANTONIO2001	9,076	8,445	65	95
SAN ANTONIO2002	8,843	8,304	62	92
SAN ANTONIO2003	10,131	9,555	76	106
SAN ANTONIO2004	8,394	7,770	63	92
SAN ANTONIO2005	10,117	9,377	75	105
SAN ANTONIO2006	9,177	8,589	70	95
SAN ANTONIO2007	9,992	9,222	74	102
SEATTLE1991	9,262	9,175	43	87
SEATTLE1992	9,687	9,462	52	91
SEATTLE1993	10,788	10,181	65	101
SEATTLE1994	9,161	8,414	65	87
SEATTLE1995	9,444	8,748	58	86
SEATTLE1996	8,955	8,324	68	87
SEATTLE1997	9,438	8,751	63	93
SEATTLE1998	9,198	8,634	65	92
SEATTLE1999	4,743	4,797	25	50
SEATTLE2000	8,591	8,519	47	87
SEATTLE2001	7,978	7,976	44	82
SEATTLE2002	8,445	8,248	47	87
SEATTLE2003	7,555	7,565	40	82
SEATTLE2004	7,964	8,016	37	82
SEATTLE2005	9,197	9,028	58	93
SEATTLE2006	8,411	8,659	35	82
SEATTLE2007	8,130	8,367	31	82
TORONTO1996	6,995	7,524	17	72
TORONTO1997	7,101	7,310	28	74
TORONTO1998	7,781	8,541	16	82
TORONTO1999	4,557	4,639	23	50
TORONTO2000	8,219	8,244	45	85
TORONTO2001	9,113	8,917	53	94
TORONTO2002	7,913	7,963	44	87
TORONTO2003	7,453	7,934	24	82
TORONTO2004	7,006	7,253	33	82
TORONTO2005	8,178	8,311	33	82
TORONTO2006	8,286	8,532	27	82
TORONTO2007	8,702	8,653	49	88
UTAH1991	9,473	9,180	58	91
UTAH1992	10,523	9,993	64	98
UTAH1993	9,145	8,988	49	87
UTAH1994	9,872	9,500	61	98
UTAH1995	9,246	8,609	62	87
UTAH1996	9,321	8,705	59	92
UTAH1997	9,466	8,721	72	92
UTAH1998	10,058	9,480	75	102
UTAH1999	5,647	5,301	42	61
UTAH2000	8,797	8,480	59	92
UTAH2001	8,407	8,043	55	87
UTAH2002	8,223	8,154	45	86
UTAH2003	8,227	8,084	48	87
UTAH2004	7,271	7,371	42	82
UTAH2005	7,625	7,975	26	82
UTAH2006	7,573	7,789	41	82
UTAH2007	9,986	9,736	60	99
WASHINGTON1991	8,313	8,721	30	82
WASHINGTON1992	8,395	8,761	25	82
WASHINGTON1993	8,353	8,930	22	82
WASHINGTON1994	8,229	8,834	24	82
WASHINGTON1995	8,242	8,701	21	82
WASHINGTON1996	6,770	6,781	28	66
WASHINGTON1997	8,215	8,138	42	83
WASHINGTON1998	7,969	7,921	42	82
WASHINGTON1999	4,560	4,672	18	50
WASHINGTON2000	7,921	8,190	29	82
WASHINGTON2001	7,645	8,192	19	82
WASHINGTON2002	7,609	7,724	37	82
WASHINGTON2003	7,502	7,585	37	82
WASHINGTON2004	7,527	7,990	25	82
WASHINGTON2005	9,245	9,297	49	92
WASHINGTON2006	8,946	8,793	44	88
WASHINGTON2007	8,922	8,999	41	86

The goal is to select an exponent, n, for the modified Pythagorean expectation equation

Expected NBA Wins ≈ Games Played * PointsFor^n/(PointsFor^n + PointsAgainst^n)

that best "fits" the actual data set. You should note that by dividing through by "Games Played" we then have an estimator of the probability of a team winning a particular game given the expected number of points scored for and against.

When attempting to match actual frequencies with predicted expectations, one frequently uses what's known as "logistic modeling". Realize that probabilities are necessarily bounded by 0 and 1 and are inherently nonlinear in nature. This can be intuitively understood by considering the difference between a 100% probability event overvalued at 99.01% (US: -10,000) versus a 50% probability event overvalued at 49.50% (US: +102). Both events are overvalued by 1%, but in the former case a growth-maximizing investor should be willing to bet and borrow every penny possibly available to him, while in the latter case such a bettor would invest less than 1% of his bankroll.

Logistic modeling defines the following function of a probability, p:

logit(p) = ln(p) - ln(1-p)

Another way of saying this is that the logit() of a probability is the logarithm of the "fair" fractional odds (i.e., decimal odds-1) associated with that probability. So in other words, the logit() of an event that occurs exactly half of the time (p=50%) would be 0, the logit() of an event that always occurs (p=100%) would be +∞, and the logit() of an event that never occurs (p=0%) would be -∞.

We then proceed by determining the exponent that minimizes the sum of the squared deviations between the logit of the actual win percentage and logit of the Pythagorean expectation. This is a straightforward problem of convex optimization well within the capabilities of Microsoft Excel Solver. If you look at Sheet1 of the attached spreadsheet you'll see that the exponent value that provides the best fit for the data set is roughly 14.02. Ths is frighteningly close to the value of 14 attributed to Dean Oliver by Wikipedia.

Determination of relative goodness-of-fit for this and other sports are left as an exercise for the interested reader (translation: don't expect me to chime in, but please see this paper if particularly motivated).

So we have:

NBA Win Probability ≈ PointsFor^14.02/(PointsFor^14.02 + PointsAgainst^14.02)

Because the OP wasn't so kind as to provide values for the variables w, x, y, and z I'll arbitrarily assign them as follows:

w = 74
x = 73
y = 68
z = 72

Using the Pythagorean expectation formula above we come up with predicted win probabilities as follows (see Sheet2 of the attached spreadsheet):

Team A: 68.44%
Team A opp: 54.98%
Team B: 59.81%
Team B opp: 64.31%

If we assume that Team A & B's opponents have themselves established their win records against "average" opposition we can then impute Team A and B's respective performances against average opposition. See the following post I had made at another site for a brief discussion of the general methodology:

I see on nba.com in the standings section that Boston has a home win percentage of 0.87879 (29-4), and that Utah has a road win percentage of 0.42857 (15-20). The question is, what is the expected probability based on these two numbers only that Boston will win tonight's game?

a) 0.5 - 0.42857 = 0.07143 + 0.87879 ==> 0.95022
b) 0.42857 + 0.87879 / 2 ==> 0.65368
c) (0.87879 - 0.42857) * 2 ==> 0.90044
d) none of the above

I not sure of the correct answer, so please explain. Thx, TTW

The Sabermetrics guys have a methodology for handling this called "log5", but it assumes a league average home team win probability of 50%.

Just thinking about it logically, here's how I'd do it:

Let A = Team A home win % (taken as an unbiased estimator of Boston's "true" home win probability versus an "average" opponent) = 29/33
Let B = Team B road win% (taken as an unbiased estimator of Utah's "true" road win probability versus an "average" opponent) = 15/35
Let H* = expected home win probability between two equally matched teams = 61% (which is roughly the total NBA home win win frequency over the past 25 years)
Let P = expected probability of A beating B playing at A's Home

Define the logit function of a probability p as the log of the "fair" fractional payout odds p/(1-p) so that:

logit(p) = ln(p) - ln(1-p)

inverted:

p = exp(logit(p))/(1+exp(logit(p)))

This gives us:

logit(A) = ln(29) - ln(4) ≈ 1.981001469
logit(B) = ln(15) - ln(20) ≈ -0.287682072
logit(H*) = ln(61%) - ln(39%) ≈ 0.447312218

So from Bayes' Theorem:

logit(P) = logit(A) - logit(B) - logit(H*)
logit(P) ≈ 1.981001469 - -0.287682072 - 0.447312218 ≈ 1.821371323

Solving for P we have:

exp(logit(P)) ≈ exp(1.246007179) ≈ 6.180327869
P = 6.180327869 / (1+6.180327869) ≈ 86.073%

So based on the above data, Boston's win probability over Utah was about 86.073%, implying fair US payout odds of about -618.

(You can also see a preview of the following not-yet-ready-for-prime-time version of a calculator that performs these calculations here. Final version to be released soon.)

So we have:

logit(A win prob vs. Avg. Opp) = logit(68.44%) - logit(54.98%) = 0.974158898
logit(B win prob vs. Avg. Opp) = logit(59.81%) - logit(64.31%) = 0.98630472

This then yields:

logit(A win prob vs. B) = logit(A win prob vs. Avg. Opp) - logit(B win prob vs. Avg. Opp) = 0.974158898 - 0.98630472 = -0.012145822

Inverting the logit function yields what's known as the logistic function:

p = 1 / (1+exp(-logit(p))

giving us a final value for the probability of A defeating B of:

p = 1 / (1+exp(0.012145822)) ≈ 49.70%

(If using the alpha-version calculator linked above, you'd enter the team in question's win probability into the "Home Team" box, 1-opponent's win probability into the "Away Team" box, and 50% into the "League Average" box. This will automatically perform the logit calculations. Comments are welcome.)

[Data]

If, however, you'd settle for roughly estimating Team A's win probability vs. Team B, then there is a straightforward (if slightly involved) methodology that may be employed to this end. Still necessary, however, is the assumption that the opponents of Team A and B have themselves played against directly relatable opposition (consider that there'd be no way one would be able to draw meaningful conclusions about the Knicks' likely performance against a Division III team based solely on the two teams' results within their respective leagues).

I think that in basketball one can get at least better than meaningless assumptions based on league averages. I would be also interested to hear from european players chiming in on eurocups linemaking.

[Ganchrow]

I think that in basketball one can get at least better than meaningless assumptions based on league averages.

Please explain how this is possible within the context of the provided example (determining fair market line between the Knicks' and a Division III team based solely on the two teams' results within their respective leagues).

Clearly there would first need to be some predefined means of making inter league comparisons. League averages would by themselves be wholly insufficient.

[Data]

Clearly there would first need to be some predefined means of making inter league comparisons.

My first thought is that in sports like basketball or baseball the game outcome, to the great extent, depends on the sum of some "units" that reflect individual players' skills, while in sports like football or soccer a team-play has much larger effect. Thus, the former sports provide the means for inter league comparisons just by adding those "units".

League averages would by themselves be wholly insufficient.

True. Using league averages, which were not mentioned in this thread, while insufficient yet necesary for making this kind of predictions, no matter intra- or inter-league.

[Ganchrow]

My first thought is that in sports like basketball or baseball the game outcome, to the great extent, depends on the sum of some "units" that reflect individual players' skills, while in sports like football or soccer a team-play has much larger effect. Thus, the former sports provide the means for inter league comparisons just by adding those "units".

True. Using league averages, which were not mentioned in this thread, while insufficient yet necesary for making this kind of predictions, no matter intra- or inter-league.

The whole point of the exercise outlined in this post is to use both Team A and B's averaging scoring (for and against), as well as that of their opponents (established versus league averages), to determine a fair money line.

[Data]

It seems that you are talking about estimating win% that is essentially can be done using variety of methods, including Pythagorean, that use offensive/defensive efficiencies (ratings, scores etc). The OP and I are talking about estimating those efficiencies/ratings/whatever.

[butters]

Perhaps this is overly simplistic, but would it be possible to estimate these values using a simple linear regression? Each previous game could be represented as:

Team_A_Off_Rating - Team_B_Def_Rating = Team A's Points For that Game (or points per 100 possessions for that game or something similar)

You could then run a regression to determine the values of each team's offensive and defensive ratings, adjusted for opponent quality. (Note that this would not produce actual points/game or efficiency values, but I think it would produce relative values that could be used to predict offensive efficiency for future matchups.) It seems to me that this would automatically account for points scored/allowed in previous games by both teams, as well as the points scored/allowed by their opponents, and their opponents' opponents, and so on...

Obviously, this wouldn't work for Ganchrow's example of the Knicks playing a DIII team, but I think that it could work for two teams that could be 'indirectly' linked through previous games.