when is a longer series BAD for the better team?
To save typing, let's define
- $p_k$ as the probability that Player A wins game $k$
- $q(j,k)$ as the probability that after $k$ games, Player A has won exactly $j$ of them
- $r_{2k-1}=\sum_{j=k}^{2k-1} q(j,k)$ is the probability that Player A has won the majority of an odd number $2k-1$ games
so we have the recurrence $$q(j,k)= p_k \,q(j-1,k-1) +(1-p_k)\,q(j,k-1)$$ starting at $q(0,0)=1$ and $q(j,0)=0$ for $j \not =0$, and given $p_1,p_2,\ldots,p_k$ we can calculate all $q(j,k)$
You seem to want
- $p_k \gt \frac12$ and $r_{2k-1} \gt \frac12$ for all positive integer $k$ since Player A is the better player
- $r_{2k-1} \gt r_{2k+1}$ for all positive integer $k$ to suggest Player A's chances get worse with a longer series
Since playing two extra games only makes a difference to the series if the score was already almost even, I think $$r_{2k-1} - r_{2k+1} = (1-p_{2k})(1-p_{2k+1})q(k,2k-1) - p_{2k} p_{2k+1} q(k-1,2k-1)$$ and so for the longer series to be worse for the better player you want this to be positive and so $$\dfrac{(1-p_{2k})(1-p_{2k+1})}{p_{2k} p_{2k+1} } > \dfrac{q(k-1,2k-1)}{q(k,2k-1)}$$
One possible approach would be to set $p_{2k+1}=p_{2k}$ and then to choose $p_{2k}$ such that $$\frac12 < p_{2k} < \dfrac{1}{1+\sqrt{\frac{q(k-1,2k-1)}{q(k,2k-1)}}}$$ where the right hand side is greater than $\frac12$ since $q(k-1,2k-1) < q(k,2k-1)$ because A is always better than B and so is always more likely to just win than to just lose
The following rather arbitrary example starts with $p_1=0.6$ and sets, using a weighted average from the inequality, $p_{2k+1}=p_{2k}=0.9\times\frac{1}{1+\sqrt{\frac{q(k-1,2k-1)}{q(k,2k-1)}}}+0.1\times\frac12$. It illustrates the point that it is not difficult to use this rule to create a long sequence, which I think meets all of your final points, and in particular the result of each game does not depend on the results of the previous games
Prob A Prob A
Game wins this winning series
game if stop here
1 0.6 0.6
2 0.5454592
3 0.5454592 0.5950459
4 0.5288103
5 0.5288103 0.5926991
6 0.5224735
7 0.5224735 0.5911743
8 0.5189403
9 0.5189403 0.5900499
10 0.5166238
11 0.5166238 0.5891616
12 0.5149601
13 0.5149601 0.5884287
14 0.5136932
15 0.5136932 0.5878058
16 0.5126882
17 0.5126882 0.5872645
18 0.5118665
19 0.5118665 0.5867864
20 0.5111790
21 0.5111790 0.5863584
22 0.5105929
23 0.5105929 0.5859713
24 0.5100859
25 0.5100859 0.5856180
26 0.5096418
27 0.5096418 0.5852932
28 0.5092487
29 0.5092487 0.5849928
30 0.5088976
31 0.5088976 0.5847133
32 0.5085817
33 0.5085817 0.5844522
34 0.5082954
35 0.5082954 0.5842072
36 0.5080346
37 0.5080346 0.5839765
38 0.5077955
39 0.5077955 0.5837585
40 0.5075755
41 0.5075755 0.5835519
42 0.5073722
43 0.5073722 0.5833557
44 0.5071834
45 0.5071834 0.5831688
46 0.5070077
47 0.5070077 0.5829904
48 0.5068436
49 0.5068436 0.5828199
50 0.5066899
51 0.5066899 0.5826564
52 0.5065455
53 0.5065455 0.5824996
54 0.5064096
55 0.5064096 0.5823489
56 0.5062814
57 0.5062814 0.5822038
58 0.5061601
59 0.5061601 0.5820640
60 0.5060452
61 0.5060452 0.5819290
62 0.5059361
63 0.5059361 0.5817986
64 0.5058324
65 0.5058324 0.5816725
66 0.5057337
67 0.5057337 0.5815504
68 0.5056395
69 0.5056395 0.5814320
70 0.5055495
71 0.5055495 0.5813172
72 0.5054635
73 0.5054635 0.5812058
74 0.5053811
75 0.5053811 0.5810975
76 0.5053022
77 0.5053022 0.5809922
78 0.5052264
79 0.5052264 0.5808897
80 0.5051536
81 0.5051536 0.5807899
82 0.5050836
83 0.5050836 0.5806927
84 0.5050162
85 0.5050162 0.5805979
86 0.5049512
87 0.5049512 0.5805054
88 0.5048886
89 0.5048886 0.5804151
90 0.5048282
91 0.5048282 0.5803269
92 0.5047698
93 0.5047698 0.5802407
94 0.5047133
95 0.5047133 0.5801565
96 0.5046587
97 0.5046587 0.5800740
98 0.5046059
99 0.5046059 0.5799934
100 0.5045547
101 0.5045547 0.5799144
My other answer had the probability of Player A winning a point not depending on the game score, but depending on the number of points played so far. The example in this answer has the conditional probability of Player A winning a point depending on the game score, but the unconditional probability staying constant over time.
This example supposes that the probability that Player A wins a point is $0.6$. But Player A is sensitive to the score, especially after an even number of games:
- If after an even number of games $2k$, Player A is losing overall then (through self-motivation) the probability that A wins the next game $2k-1$ is $1$
- If after an even number of games $2k$, scores are even then (due to Player A choking or freezing, and this gets worse over time) the probability that A wins the next game is $0.6^{k+1}$
- If after an even number of games $2k$, Player A is winning then (through relaxation) the probability that A wins the next game tends to increase and happens to be $1-\frac{0.4-\left(1- 0.6^{k+1}\right)\mathbb{P}(\text{A tying after }2k\text{ games})}{\mathbb{P}(\text{A winning after }2k\text{ games})}$
- If after an odd number of games $2k-1$, the probability that A wins the next game is $0.6$, no matter what the score is at the time
The overall effect is to keep the probability that Player A wins a particular point at $0.6$, but surprisingly the probability that Player A is ahead after an odd number of games keeps falling (i.e. would win a series of that length), below $0.5$. I would expect it to be possible to extend the example, and perhaps the limit of the probability that A wins a long series is just over $0.4729$
Games Prob A is Prob Prob A is Prob A wins Prob A wins Prob A wins
played losing series winning next game next game next game
so far series tied series if behind if tied if ahead
0 0 1 0 0.6
1 0.4 0.6 0.6 0.6
2 0.16 0.48 0.36 1 0.36 0.742222222
3 0.4672 0.5328 0.6 0.6
4 0.18688 0.38656 0.42656 1 0.216 0.772747187
5 0.48994304 0.51005696 0.6 0.6
6 0.195977216 0.351566124 0.452456660 1 0.1296 0.792252266
7 0.501980370 0.498019630 0.6 0.6
8 0.200792148 0.334642831 0.464565021 1 0.07776 0.803302033
9 0.509413153 0.490586847 0.6 0.6
10 0.203765261 0.325848417 0.470386322 1 0.046656 0.810040466
11 0.514410894 0.485589106 0.6 0.6
12 0.205764358 0.321140653 0.473094989 1 0.0279936 0.814309532
13 0.517915128 0.482084872 0.6 0.6
14 0.207166051 0.318607172 0.474226777 1 0.01679616 0.817082862
15 0.520421846 0.479578154 0.6 0.6
16 0.208168739 0.317259969 0.474571292 1 0.010077696 0.818915974
17 0.522231458 0.477768542 0.6 0.6
18 0.208892583 0.316564635 0.474542781 1 0.006046618 0.820143698
19 0.523543073 0.476456927 0.6 0.6
20 0.209417229 0.316225505 0.474357266 1 0.003627971 0.820975122
21 0.524495478 0.475504522 0.6 0.6