TITLE: when is a longer series BAD for the better team?
QUESTION [8 upvotes]: This question is inspired by: A longer series is better for a better team: Can you see this at a glance?
Also obviously inspired by the NBA playoffs happening right now.  :)
Suppose two teams are playing a series of $2k-1$ games, and the first team to win $k$ games wins the overall series.  (No game can end in a tie.)  Moreover, team A is "better" than team B.  

Let $A_i$ denote the event that team A wins game $i$.  
Let $A_{series}$ denote the event that team A wins the series, i.e., A wins $k$ or more games.

If the game results are i.i.d., and $P(A_i) = p > 1/2$ for any game $i$, then a longer series (larger $k$) increases A's chance of winning the series, i.e. $P(A_{series})$ is an increasing function in $k$.  This is intuitively obvious, and a proof can be found in the link above (although that post asks an excellent question re: why such an "obvious" result requires an algebraically convoluted proof).
I wanna know under what conditions, i.e. under what probability model, would a longer match be BAD for the better team A.  Perhaps some dependence that encodes "reversion to the mean" and/or (opposite of) "momentum"?
What I tried
My $0$-th attempt: If we allow "fatigue" in the form of decreasing $P(A_i),$ then a longer series can be bad for A, even when all $P(A_i) > 1/2.$  A simple 3-game example: if $P(A_1) = 1, P(A_2) = P(A_3) = 0.51$, then A always wins in a 1-game "series" ($k=1$) but B has a chance ($0.49^2$) in a 3-game series ($k=2$).  @Henry in his answer gave an infinite length example.  So I'm looking for something where the marginal probabilities $P(A_i)$ are constant (i.e. no fatigue).
My 1st attempt: let $P(A_1) = p > 1/2$, and thereafter every game $i$ result = game 1 result.  This means: (1) $P(A_i) = p > 1/2$ (even though they are dependent), and yet (2) a longer series would give no advantage (nor disadvantage) to team A, because $P(A_{series}) = p$ regardless of $k$.  However, I want a scenario where a longer series actually decreases $P(A_{series})$.
My 2nd attempt is something convoluted: the first 7 games ($k=4$) are played "normally" (i.i.d.) but then games 8 and 9 are always won by the loser of the best-of-first-7 series.  I have not worked this out fully, but while this may give an example where $P(A_{series} | k = 4) > P(A_{series} | k = 5)$, I think this also implies the marginal probabilities $P(A_8), P(A_9) < 1/2$.  So this isn't satisfying as a counterexample, since not only $P(A_i)$ are non-constant, team A actually becomes the worse team in a sense.
My 3rd attempt is a "surgical tweak" to the 2nd attempt: (1) if the first 7 games include exactly 4 wins by A, then games 8 & 9 are won by team B (therefore making B the overall winner), but (2) if the first 7 games have any other result, then games 8 & 9 are played i.i.d. but with an enhanced $P(A_8)=P(A_9) = p' > p$.  By restricting the special dependence to an event of small enough probability, and balancing it out with $p' > p$ in the case (2), I think I can manage $P(A_8) = P(A_9) =p $.  I have not worked out whether $P(A_{series} | k = 4) > P(A_{series} | k = 5)$ for some choice of $p, p'$. 
 Also, this counterexample is too "artificial" for my taste.  
What I seek is a probability model where:

For every game $i$, the marginal probability $P(A_i)$ is the same, i.e. $\forall i: P(A_i) = p > 1/2$.
There exists $k_0$ s.t. $P(A_{series} | k = k_0) > P(A_{series} | k = k_0 + 1)$.  

Bonus if this is true for all $k_0$, or all sufficiently large $k_0$.

Aesthetic requirement :) - Any dependence is as "simple" as possible, i.e. I prefer not to have something like my 3rd attempt (or even more convoluted).  I know this isn't a math requirement, and people can have different tastes...  comments on this point are welcome.

REPLY [3 votes]: 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