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

Tags:

Probability