Fast Track: where playing to win lost (seed 69)

The 80-game run of "own less the leader" against the champion had 9 games that red won only as the champion. This note takes one of them, seed 69, apart. A short table of the other eight is at the end.

How it was run (fasttrack/exp_pivot.roc):

Red plays seed 69 twice: once as the champion (its own pieces only) and once as the chaser. The chaser scores its own pieces less the leading opponent's, at the end of its turn.
Blue, green and purple play the champion in both games.
Each player's deck is shuffled once from the seed, so both games deal the same cards in the same order.
The two games are identical until red's two strategies first choose differently. That position is the pivot.

Board values are each player's own, as the champion values its pieces: Steve's square ranking (100 points a place) plus the base bonus of 1000 a step.

Who won

red plays as	winner	red home on
the champion	red, on turn 10	turn 10
the chaser	blue, on turn 17	turn 18

The pivot: red's turn 6

Red holds 7 2 9 6 6. Board values before the turn:

red	blue	green	purple
15600	5500	5700	18400 (the leader)

Both strategies play the same cards: a 6 brings a piece out of red's pen to L0, and then the 7 is split. The only difference is how the 7 splits:

The champion splits it 5 + 2. The piece on R0 goes 5 into base, to B3. The new piece goes 2, to L2.
The chaser splits it 2 + 5. R0 goes 2, to DS. The new piece goes 5, to red's fast-track square, where purple's piece is standing, and purple's piece goes back to its pen.

For the two players in the capture, red and purple:

	red	purple
before	15600	18400
champion's pick (B3, L2)	21100 (+5500)	18400 (0)
chaser's pick (DS, FT, capture)	20300 (+4700)	13000 (−5400)

So the capture costs red 800 of its own value and takes 5400 off purple.

How each strategy scores the two lines:

The champion scores red's own value (plus a hoarded hand, the same in both lines here): 21100 against 20300, so it tucks into B3.
The chaser subtracts the leader's value. The capture puts purple at 13000, still the leader, since blue and green are at 5500 and 5700. So it scores 21100 − 18400 = 2700 against 20300 − 13000 = 7300, and captures.

Every line red had

All 30 lines of red's turn 6, sorted by the champion's score. Each pair of rows with the same squares is one position the search failed to merge; that is a separate issue, covered at the end.

line	pieces	red	purple (leader)	others	champion's score	chaser's score
champion's pick	red@red.R0 red@red.HP2 -> red@red.B3 red@red.L2	+5500	0		21100	2700
	red@red.R0 red@red.HP2 -> red@red.L2 red@red.B3	+5500	0		21100	2700
	red@red.R0 red@red.HP2 -> red@red.B1 red@red.L4	+5200	0		20800	2400
	red@red.R0 red@red.HP2 -> red@red.L4 red@red.B1	+5200	0		20800	2400
	red@red.R0 red@red.HP2 -> red@red.B2 red@red.L3	+4700	0		20300	1900
	red@red.R0 red@red.HP2 -> red@red.L3 red@red.B2	+4700	0		20300	1900
chaser's pick	purple@red.FT red@red.R0 red@red.HP2 -> red@red.FT purple@purple.HP2 red@red.DS	+4700	−5400		20300	7300
	red@red.R0 red@red.HP2 -> red@red.B3 red@blue.R2	+4700	0		20300	1900
	red@red.R0 red@red.HP2 -> red@blue.R2 red@red.B3	+4700	0		20300	1900
	red@red.R0 red@red.HP2 -> red@red.B2 red@blue.R1	+4600	0		20200	1800
	red@red.R0 red@red.HP2 -> red@blue.R1 red@red.B2	+4600	0		20200	1800
	red@red.R0 red@red.HP2 -> red@bullseye red@red.BR	+4300	0		19900	1500
	red@red.R0 red@red.HP2 -> red@red.DS red@bullseye	+4200	0		19800	1400
	blue@blue.R0 red@red.R0 red@red.HP2 -> red@red.B1 red@blue.R0 blue@blue.HP4	+3100	0	blue −5500	18700	300
	blue@blue.R0 red@red.R0 red@red.HP2 -> red@blue.R0 blue@blue.HP4 red@red.B1	+3100	0	blue −5500	18700	300
	red@red.R0 red@red.HP2 -> red@red.DS red@red.L0	+2900	0		18500	100
	red@red.HP2 -> red@blue.R1	+2200	0		17800	−600
	red@red.HP2 -> red@red.L2	+2000	0		17600	−800
	red@red.R0 red@red.HP2 -> red@red.DS red@blue.BR	+1700	0		17300	−1100
	red@red.R0 red@red.HP2 -> red@blue.BR red@red.DS	+1700	0		17300	−1100
	red@red.R0 red@red.HP2 -> red@red.BR red@blue.DS	+1500	0		17100	−1300
	red@red.R0 red@red.HP2 -> red@blue.DS red@red.BR	+1500	0		17100	−1300
	red@red.HP2 -> red@blue.R2	+1200	0		16800	−1600
	red@red.HP2 -> red@blue.HH	+1000	0		16600	−1800
	red@red.HP2 -> red@blue.L1	+700	0		16300	−2100
	red@red.HP2 -> red@blue.R3	+300	0		15900	−2500
	red@red.R0 red@red.HP2 -> red@red.BR red@blue.R4	+300	0		15900	−2500
	red@red.R0 red@red.HP2 -> red@blue.R4 red@red.BR	+300	0		15900	−2500
	red@red.R0 red@red.HP2 -> red@red.DS red@blue.R4	+200	0		15800	−2600
	red@red.R0 -> red@red.DS	+100	0		15700	−2700

One line captures blue instead, costing blue 5500. The chaser scores it only 300: blue is not the leader, so taking blue's piece is worth nothing to it.

What happened next

Board values at the start of each of red's turns, and every capture.

Red as the champion:

red's turn	red	blue	green	purple
7	21100	15600	12900	19500
8	26500	19100	16800	20100	red captures purple
9	28700	19100	19500	19100
10	32500	19100	23000	19100	red home: red wins

Red as the chaser:

red's turn	red	blue	green	purple
7	20300	15600	12900	12400
8	24900	19100	16800	12000	purple captures red
9	21900	19100	19500	14200
10	27400	19100	23000	18000
11	28700	21900	23200	19100
12	26500	24500	23200	19100
13	29400	27000	25500	19100	green captures red
14	27000	29300	29200	21900
15	27000	31600	31800	23700
16	27000	32700	31600	27000
17	29800	32200	32600	27000	blue home: blue wins
18	30900	33800	32400	31600	red home

Reading it

The capture came instead of a piece tucked into base. The champion put a piece on B3, where no one can touch it. The chaser left red's pieces on DS and the fast-track square. Purple's pieces pass through red's zone on their way home.
The capture was paid back almost at once. Two turns later purple captured red. Red went from 24900 to 21900, and red's lead over the field was gone by turn 14. Green captured red again on turn 13, and red sat at 27000 for three turns while blue and green went past.
The champion took purple anyway, and for free. On turn 8, as the champion, red captured purple in passing, as part of its own best play. The chaser paid for a capture the champion got two turns later at no cost.
The capture did not stop the leader from winning, because the leader changed. Purple was the leader at the pivot; it never threatened again. Blue, 5500 at the pivot and ignored by the chaser, won.

A caveat on cause. After the pivot the two games differ, and every later move is the champion's best play in a different position. The cards come in the same order, but they meet different boards. So the rest of each game is not caused by the pivot alone. What the pivot shows cleanly is the trade: red gave up a safe B3 for a capture that left two pieces exposed in an opponent's path.

The other eight games

The first point where red's two strategies part, and who won each game.

seed	red's turn	champion's pick	chaser's pick (red / the leader)	winner, red as champion	winner, red as chaser
74	3	a piece to B1, keeping the A (+1400)	captures green's only piece out, at green's DS (+2800 / −5600), spending the hoarded A	red, turn 10	green, turn 14
21	8	a piece to B2 (+3700)	captures blue on blue's R0 (+1800 / −5500)	red, turn 12	purple, turn 15
22	2	B1 and a piece out (+8400)	captures blue at BR, and a J trade sends green back (+5100 / −5700)	red, turn 10	purple, turn 18
43	3	a piece to R3 (−300)	green's piece to its pen with no red piece moving (0 / −2800)	red, turn 16	purple, turn 18
55	2	out, to FT (+4600)	out, onto purple at the bullseye (+4100 / −4100)	red, turn 14	green, turn 17
60	4	out, to R0 (+5500)	captures blue on blue's FT (+3100 / −4600)	red, turn 10	blue, turn 13
76	6	B2, capturing purple on the way (+3800)	captures blue on blue's R0 (−1000 / −5500)	red, turn 14	blue, turn 15
79	2	a piece to R3 (−800)	J trade with green's piece at the bullseye (−1600 / −2600)	red, turn 12	green, turn 15

The same pattern shows up in most of them: a base square or a safe advance given up for a capture of the leader. The winner is usually a player the chaser was not chasing at the pivot: in 5 of these 9 games someone other than the pivot's leader won.

Seed 43 looks wrong. The chaser's line sends green's piece home while every red piece ends where it started. With K J J 3 3 there might be a legal sequence, a J trade and back, but I have not worked it out. It is worth checking before trusting anything that involves the J.

The search does not merge identical positions. The seed 69 table lists the same squares twice in reverse order: red to B3 and L2, then red to L2 and B3. The search merges lines by comparing the whole game record, and these two probably differ only in the order the pieces are listed. If so, it changes no decision but multiplies the work. Seed 74 had 1,257 lines. This is the first thing to measure for speed.