Ask The Wizard #411

No, I disagree.

Effective money management in gambling is often exaggerated. Regardless of how a player approaches a game of chance like roulette, the house edge will remain intact.

Regarding the idea that casinos can simply outlast players until they run out of funds, it's also plausible that the casino might end up facing bankruptcy first. For instance, consider a player with a bankroll of $10,000 competing against an MGM-Mirage casino. At the time of writing, MGM-Mirage has a market capitalization of $11,168,000,000 . Assuming for argument’s sake that the company owns all its shares, if the player continues betting until one of them runs out of money, they would have a 1 in 1,116,801 chance of winning the entire casino. Given the high volume of players, it’s likely that eventually, one will triumph.

In essence, every state is guaranteed at least one Congressional seat. Subsequently, additional seats are allocated sequentially to the states that would benefit the most from them.

The process of determining which state receives the 'greatest benefit' is quite complex. Here are the specifics:

1. Calculate the \"multiplier\" as the geometric mean based on (1) the percentage increase in overall seats once an extra seat is added and (2) the percentage increase in total seats if the state had one additional seat more than it currently possesses. To articulate it mathematically, if n represents the existing number of seats, the multiplier can be expressed as 1/sqrt(n*(n+1)).
2. To derive the 'priority value', multiply the previously calculated multiplier by the state's population.

Perform this computation for all states. The state with the highest priority value is awarded the next available seat.

For instance, consider Nevada. The population recorded in 2020 was 3,104,614, and it has 4 seats already. The multiplier is calculated as 1/sqrt(4*5) = 1/sqrt(20) = 0.223607. Thus, the priority value amounts to 3,104,614 * 0.223607 = 694,213.

You may not have asked, but if Congress were to add another seat, the 436th, it would be allocated to New York. The table below displays the priority values for all 50 states ranked accordingly.

State	Population	Current Seats	Multiplier	Priority Value
New York	20,201,249	26	0.037743	762,447
Ohio	11,799,448	15	0.064550	761,651
Texas	29,145,505	38	0.025976	757,090
Florida	21,538,187	28	0.035093	755,842
Arizona	7,151,502	9	0.105409	753,835
California	39,538,223	52	0.019048	753,143
Virginia	8,631,393	11	0.087039	751,266
Idaho	1,839,106	2	0.408248	750,812
Michigan	10,077,331	13	0.074125	746,981
New Jersey	9,288,994	12	0.080064	743,715
Pennsylvania	13,002,700	17	0.057166	743,315
Massachusetts	7,029,917	9	0.105409	741,018
Georgia	10,711,908	14	0.069007	739,192
Washington	7,705,281	10	0.095346	734,670
Illinois	12,812,508	17	0.057166	732,442
West Virginia	1,793,716	2	0.408248	732,281
Utah	3,271,616	4	0.223607	731,556
Tennessee	6,910,840	9	0.105409	728,466
Maryland	6,177,224	8	0.117851	727,993
Missouri	6,154,913	8	0.117851	725,363
Oklahoma	3,959,353	5	0.182574	722,876
North Carolina	10,439,388	14	0.069007	720,386
Louisiana	4,657,757	6	0.154303	718,708
Indiana	6,785,528	9	0.105409	715,257
Iowa	3,190,369	4	0.223607	713,388
Delaware	989,948	1	0.707107	699,999
Kentucky	4,505,836	6	0.154303	695,266
Wisconsin	5,893,718	8	0.117851	694,581
Nevada	3,104,614	4	0.223607	694,213
South Carolina	5,118,425	7	0.133631	683,978
Colorado	5,773,714	8	0.117851	680,439
Arkansas	3,011,524	4	0.223607	673,397
Minnesota	5,706,494	8	0.117851	672,517
Alabama	5,024,279	7	0.133631	671,398
Mississippi	2,961,279	4	0.223607	662,162
Connecticut	3,605,944	5	0.182574	658,352
Kansas	2,937,880	4	0.223607	656,930
Oregon	4,237,256	6	0.154303	653,823
South Dakota	886,667	1	0.707107	626,968
New Mexico	2,117,522	3	0.288675	611,276
Hawaii	1,455,271	2	0.408248	594,112
Nebraska	1,961,504	3	0.288675	566,237
New Hampshire	1,377,529	2	0.408248	562,374
Maine	1,362,359	2	0.408248	556,181
North Dakota	779,094	1	0.707107	550,903
Alaska	733,391	1	0.707107	518,586
Vermont	643,077	1	0.707107	454,724
Rhode Island	1,097,379	2	0.408248	448,003
Montana	1,084,225	2	0.408248	442,633
Wyoming	576,851	1	0.707107	407,895
Total	330,759,736	435	0.002296	759,495

To address your second question, yes, this scenario can indeed occur. I had to spend some time examining potential cases to find one that fits. Imagine a fictional country, consisting of six states, with a total of 15 seats allocated as follows:

State	Population	Current Seats	Multiplier	Priority Value
Confusion	900	3	0.288675	259.807621
Denial	800	3	0.288675	230.940108
Bliss	700	3	0.288675	202.072594
Depression	600	2	0.408248	244.948974
Limbo	500	2	0.408248	204.124145
Anxiety	400	2	0.408248	163.299316
Total	3,900	15

Now, let’s consider a population census where numbers change substantially, yet the total population remains 3,900, with Anxiety's population increasing from 400 to 401. Despite this change, it would still be entitled to only one seat among 15 states.

State	Population	Current Seats	Multiplier	Priority Value
Confusion	1,885	7	0.133631	251.893721
Denial	405	2	0.408248	165.340558
Bliss	404	2	0.408248	164.932309
Depression	403	2	0.408248	164.524061
Limbo	402	1	0.707107	284.256926
Anxiety	401	1	0.707107	283.549819
Total	3,900	15

I want to reiterate that I had to adjust the figures considerably to discover such a situation. In general, the allocation method appears quite just.

Here is my solution (PDF).

This topic is being explored and discussed on my forum at Wizard of Vegas .