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00:00:03,280 --> 00:00:05,470
Okay, so we've seen that propositional
nets can

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00:00:05,470 --> 00:00:07,530
be used to advantage in searching game
trees.

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00:00:08,550 --> 00:00:13,300
The other main use of propnets is in
analyzing games.

4
00:00:13,300 --> 00:00:15,380
For example, by analyzing the propnet for
a

5
00:00:15,380 --> 00:00:17,210
game, it's sometimes possible for a player
to

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00:00:17,210 --> 00:00:19,410
detect structure that allows him to
actually decrease

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00:00:19,410 --> 00:00:21,300
the size of the game tree in significant
ways.

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00:00:22,580 --> 00:00:26,840
Game decomposition, or so-called
factoring, is a good example of this.

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00:00:30,250 --> 00:00:32,770
A compound game is a single game
consisting

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00:00:32,770 --> 00:00:36,220
of two or more individual games glued
together.

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00:00:36,220 --> 00:00:37,600
The state of the compound game is a

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00:00:37,600 --> 00:00:40,620
composition of the states of the
individual games.

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00:00:40,620 --> 00:00:42,670
On each step of a compound game the

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00:00:42,670 --> 00:00:45,400
players perform actions in each of the
individual games.

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00:00:46,750 --> 00:00:50,760
The compound game is over when either one
or all of the individual games are over.

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00:00:50,760 --> 00:00:52,180
Depending on the type of the compound
game.

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00:00:55,210 --> 00:00:56,890
Now using the techniques we have seen thus

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00:00:56,890 --> 00:01:00,290
far, compound games can be quite expensive
to play.

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00:01:00,290 --> 00:01:05,190
Unless a player recognizes that there are
independent subgames involved, it's likely

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00:01:05,190 --> 00:01:07,920
to search a game tree that's far larger
than it needs to be.

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00:01:09,220 --> 00:01:12,320
For example, if one subgame has branching
factor a,

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00:01:12,320 --> 00:01:15,090
and a second has branching factor b, then
the

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00:01:15,090 --> 00:01:17,410
branching factor of the joint game is a
times

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00:01:17,410 --> 00:01:20,920
b, one choice of each of the b actions

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00:01:20,920 --> 00:01:23,010
for each of the a actions.

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00:01:23,010 --> 00:01:25,320
Then the fringe of the game tree at depth
d is likely

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00:01:25,320 --> 00:01:28,700
to be something like (a x b) raised to the
power d.

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00:01:31,740 --> 00:01:33,940
Now this is wasteful if the two games are

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00:01:33,940 --> 00:01:37,660
truly independent, in that case there are
two trees, one

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00:01:37,660 --> 00:01:40,420
with branching factor a and one with
branching factor b

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00:01:40,420 --> 00:01:43,190
and the total size of the fringe of these
trees.

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00:01:43,190 --> 00:01:48,650
At depth d, should be only a to the d plus
b to the d, significantly smaller.

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00:01:51,320 --> 00:01:53,440
Double Tic Tac Toe is an example.

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00:01:53,440 --> 00:01:57,170
On each move, players play on one of the
two boards.

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00:01:57,170 --> 00:01:59,080
First to win on either board wins the
overall game.

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00:02:00,130 --> 00:02:03,320
If the game is left unfactored, the
branching factors are quite large.

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00:02:03,320 --> 00:02:07,120
There are 81 possibilities in that first
move.

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00:02:07,120 --> 00:02:11,220
Nine for each board, and then they're 64
in a second and so forth.

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00:02:11,220 --> 00:02:14,430
However, the games are factored, the
branching factors are much more modest.

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00:02:17,870 --> 00:02:19,300
Game factoring is the process of

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00:02:19,300 --> 00:02:23,250
discovering independent games inside of
larger games.

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00:02:23,250 --> 00:02:25,620
Once discovered, game players can use
these factors

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00:02:25,620 --> 00:02:28,520
to play the individual games independently
of each other.

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00:02:28,520 --> 00:02:32,109
And those cut down on the combinatory cost
of playing such games.

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00:02:32,109 --> 00:02:35,390
Turns out that it's often easier to
discover independent subgames

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00:02:35,390 --> 00:02:39,790
using the propnet representation for
games, rather than other representations.

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00:02:42,490 --> 00:02:45,160
Now in this segment, we'll look at

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00:02:45,160 --> 00:02:48,570
some elementary techniques for factoring
games using propnets.

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00:02:48,570 --> 00:02:52,450
We'll first talk about discovering factors
with completely independent subgames.

50
00:02:53,450 --> 00:02:57,630
We'll then talk about factoring with
interdependent termination and rewards.

51
00:02:57,630 --> 00:03:01,599
And after that, we talk about factoring
with interdependent actions as well.

52
00:03:02,880 --> 00:03:06,230
And finally we'll discuss briefly the
conditional factors.

53
00:03:06,230 --> 00:03:07,670
That is factors that appear only

54
00:03:07,670 --> 00:03:09,950
in certain states of the games but not in
all states.

55
00:03:12,960 --> 00:03:14,650
Okay, so let's begin our discussion of
factoring

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00:03:14,650 --> 00:03:16,970
with a simple case of a compound game.

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00:03:18,430 --> 00:03:23,120
consisting of multiple and completely
independent subgames.

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00:03:23,120 --> 00:03:26,810
Only one of which is relevant to the
overall game.

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00:03:26,810 --> 00:03:29,630
Without factoring, the player is likely to
consider actions of

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00:03:29,630 --> 00:03:33,930
all of the subgames, when only one of
those subgames matters.

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00:03:33,930 --> 00:03:36,730
Now, okay, now this is admittedly a very
special case.

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00:03:36,730 --> 00:03:37,850
It doesn't arise often,

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00:03:37,850 --> 00:03:40,780
and it can be solved by means other than
factoring.

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00:03:40,780 --> 00:03:43,110
But not by the method we've seen thus far.

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00:03:43,110 --> 00:03:45,514
Nevertheless, it's worth considering
because of it prepares

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00:03:45,514 --> 00:03:47,500
us for factoring more complicated sorts of
games.

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00:03:49,450 --> 00:03:52,960
Okay, Multiple Buttons and Lights is an
example of a game of this sort.

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00:03:52,960 --> 00:03:57,690
The overall game consists of multiple
copies of mul-, buttons and lights, and

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00:03:57,690 --> 00:04:02,070
each copy of buttons and lights, there are
three base propositions for lights.

70
00:04:02,070 --> 00:04:03,310
And three actions,

71
00:04:03,310 --> 00:04:04,620
the buttons.

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00:04:04,620 --> 00:04:08,820
Pushing the first button in each group
toggles the first light in that group.

73
00:04:08,820 --> 00:04:11,160
Pushing the second button in each group
interchanges

74
00:04:11,160 --> 00:04:13,450
the first and second lights in that group.

75
00:04:13,450 --> 00:04:15,340
And pushing the third button in each group

76
00:04:15,340 --> 00:04:18,000
interchanges the second and third lights
in that group.

77
00:04:19,080 --> 00:04:21,430
Initially, all of the lights are off.

78
00:04:21,430 --> 00:04:25,160
The goal is to turn on all of the lights
in the middle group.

79
00:04:25,160 --> 00:04:26,640
The setting of the others doesn't matter.

80
00:04:30,030 --> 00:04:32,220
Now here's a propnet for the game.

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00:04:32,220 --> 00:04:34,720
Notice that there are three disjoined
parts of the propnet.

82
00:04:34,720 --> 00:04:37,650
One portion for the first group of buttons
and lights, a second

83
00:04:37,650 --> 00:04:40,450
portion for the second group, and a third
portion for the third group.

84
00:04:40,450 --> 00:04:44,210
And note that the goal and termination
conditions for the overall

85
00:04:44,210 --> 00:04:47,130
game are based entirely on the lights in
the second group.

86
00:04:49,430 --> 00:04:51,010
Looking at the propnet for this game, it's
easy to

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00:04:51,010 --> 00:04:53,100
see that the game is of this very special
structure.

88
00:04:54,122 --> 00:04:55,830
The propnet consists entirely of three

89
00:04:55,830 --> 00:04:58,820
completely disconnected subnets, one for
each subgame.

90
00:04:58,820 --> 00:05:01,400
Finding factors in situations like this is
really easy, it can

91
00:05:01,400 --> 00:05:04,030
be computed in time that's polynomial in
the size of the propnet.

92
00:05:04,030 --> 00:05:07,100
And having found these factors, the
solution is really simple.

93
00:05:07,100 --> 00:05:09,770
The player just keeps the essential
component and discards the other two.

94
00:05:11,590 --> 00:05:12,580
Now, note that this technique can be

95
00:05:12,580 --> 00:05:15,450
applied equally well to multiple player
games.

96
00:05:15,450 --> 00:05:17,606
For example, Multiple Tic Tac Toe, that is
three games of

97
00:05:17,606 --> 00:05:21,160
Tic Tac Toe glued together, in which only
the middle game matters.

98
00:05:21,160 --> 00:05:22,899
Propnet from Multiple Tic Tac Toe is
similar

99
00:05:22,899 --> 00:05:24,930
to the propnet from Multiple Buttons and
Lights.

100
00:05:24,930 --> 00:05:27,490
And it's possible to find the structure in
Multiple Tic

101
00:05:27,490 --> 00:05:30,310
Tac Toe just as easily as from Multiple
Buttons and Lights.

102
00:05:33,300 --> 00:05:36,800
Okay, now let's consider a slightly more
complicated case.

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00:05:36,800 --> 00:05:41,040
Namely, compound games with interdependent
termination.

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00:05:42,340 --> 00:05:45,230
We can assume interdependent goals as well
but I'm going

105
00:05:45,230 --> 00:05:48,240
to ignore goals for now and just focus on
termination.

106
00:05:49,620 --> 00:05:53,708
As with the case of independent subgames,
entirely independent subgames actions are

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00:05:53,708 --> 00:05:55,836
petitioned over distinct subgames and
there

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00:05:55,836 --> 00:05:58,523
are no incoming connections between those
subgames.

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00:05:58,523 --> 00:06:02,240
The main difference is that the
termination condition for the overall game

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00:06:02,240 --> 00:06:05,290
can depend on more than one and perhaps
all of the subgames.

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00:06:06,790 --> 00:06:10,170
Now in games of this sort, the termination
of the overall game

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00:06:10,170 --> 00:06:14,500
can be defined as any boolean combination
of conditions in the individual games.

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00:06:14,500 --> 00:06:16,647
In the case where the combinations are
disjunction

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00:06:16,647 --> 00:06:19,060
then the game is said to be disjunctive.

115
00:06:19,060 --> 00:06:20,470
In the case where the combinations is

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00:06:20,470 --> 00:06:22,449
conjunct the game is said to be
conjunctive.

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00:06:24,540 --> 00:06:26,330
As a simple example of a factorable game

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00:06:26,330 --> 00:06:31,910
with disjunctive termination, consider
Best Buttons and Lights.

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00:06:31,910 --> 00:06:35,310
In multiple buttons and lights, only one
group of buttons and lights matters.

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00:06:35,310 --> 00:06:37,960
In Best Buttons and Lights, the compound
games

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00:06:37,960 --> 00:06:42,990
terminate, compound game terminates,
whenever any group terminates.

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00:06:42,990 --> 00:06:44,450
Not just the one in the middle.

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00:06:44,450 --> 00:06:47,220
This gives the player the freedom to play
whichever subgame

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00:06:47,220 --> 00:06:49,650
it likes and rest assured that if it
succeeds on

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00:06:49,650 --> 00:06:53,376
that subgame, it succeeds on the overall
game with the same score.

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00:06:53,376 --> 00:06:58,105
Okay here's a propnet for Best Buttons and
Lights.

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00:06:58,105 --> 00:07:02,620
It's similar to the propnet for Multiple
Buttons and Lights, except that the goals

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00:07:02,620 --> 00:07:09,070
and termination are based disjunctively on
all three subgames.

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00:07:09,070 --> 00:07:10,750
The good news is that we can

130
00:07:10,750 --> 00:07:15,980
extend the technique, this discussed, in
just discussed.

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00:07:15,980 --> 00:07:18,290
Let's consider the simply disjunctive
case.

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00:07:18,290 --> 00:07:21,400
If the connective leading to a termination
is a disjunction as it is

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00:07:21,400 --> 00:07:25,680
in this case with its inputs, in terms
supplied by nodes in different subgames,

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00:07:25,680 --> 00:07:34,200
and we simply cut off the overall
termination condition, and and, and make

135
00:07:34,200 --> 00:07:36,235
the termination of each of the individual

136
00:07:36,235 --> 00:07:39,020
subnets a termination condition for that
subnet.

137
00:07:39,020 --> 00:07:41,000
And we

138
00:07:41,000 --> 00:07:43,550
can repeat this process as long as we only
encounter disjunctions.

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00:07:43,550 --> 00:07:46,570
If the game is truly disjunctive, this
will eventually

140
00:07:46,570 --> 00:07:48,880
lead to a separable propnet like the ones
shown here.

141
00:07:51,260 --> 00:07:52,880
Now, if the process does succeed in

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00:07:52,880 --> 00:07:54,440
factoring the propnet in this way, then
the

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00:07:54,440 --> 00:07:56,600
player simply picks one of the subgames,
and

144
00:07:56,600 --> 00:08:00,510
proceeds as with the case of independent
subgames.

145
00:08:00,510 --> 00:08:03,086
Alternatively and better, the player tries
playing all of

146
00:08:03,086 --> 00:08:05,600
the subgames, and picks the one with the
best score.

147
00:08:05,600 --> 00:08:06,900
Or, at least that's the basic idea.

148
00:08:08,940 --> 00:08:11,170
Unfortunately, it's not quite that simple.

149
00:08:11,170 --> 00:08:13,360
There's a problem that does not arise

150
00:08:13,360 --> 00:08:16,040
in the case of completely independent
subgames.

151
00:08:16,040 --> 00:08:18,340
The player may choose one subgame, find

152
00:08:18,340 --> 00:08:20,910
a winning strategy, and begin executing
that strategy.

153
00:08:21,960 --> 00:08:24,220
Unfortunately in the course of execution
on the

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00:08:24,220 --> 00:08:28,590
chosen subgame, one of the other subgames
may terminate.

155
00:08:28,590 --> 00:08:30,230
Terminating the game as a whole before

156
00:08:30,230 --> 00:08:32,970
the strategy in the chosen subgame is
complete.

157
00:08:32,970 --> 00:08:33,960
This can lead to a lower

158
00:08:33,960 --> 00:08:35,890
score than a player might have otherwise
expected.

159
00:08:37,140 --> 00:08:39,954
The solution to this problem is to check
each of the

160
00:08:39,954 --> 00:08:44,890
subgames for terminations when no actions
are played in that subgame.

161
00:08:44,890 --> 00:08:46,830
The player takes the shortest time period
and

162
00:08:46,830 --> 00:08:48,860
plays each of the other subgames with that

163
00:08:48,860 --> 00:08:50,800
as a step limit and takes the one

164
00:08:50,800 --> 00:08:53,106
that provides the best result within that
step limit.

165
00:08:53,106 --> 00:08:56,600
For the subgame with the shortest
termination, if there are no

166
00:08:56,600 --> 00:08:59,160
other subgames with that step limit, the
player tries the next

167
00:08:59,160 --> 00:09:00,720
shortest step limit and so forth.

168
00:09:02,220 --> 00:09:05,610
Although in this approach the player
searches all of the sub-games more than

169
00:09:05,610 --> 00:09:09,750
once, this is usually a lot less expensive
that searching the game tree with

170
00:09:09,750 --> 00:09:13,550
the unfactored game because it's not, not
cross multiply the branching factors of

171
00:09:13,550 --> 00:09:17,160
the independent games as it would if it
did not use the game's factors.

172
00:09:18,410 --> 00:09:20,235
And once again, it's possible to extend
this technique to

173
00:09:20,235 --> 00:09:22,460
multi-player games, but this has to be
done with some care.

174
00:09:25,410 --> 00:09:27,490
Okay, finally, let's look at compound

175
00:09:27,490 --> 00:09:33,260
games with interdependent actions,
interdependent, interdependent actions.

176
00:09:33,260 --> 00:09:37,300
By interdependence of actions, we mean
those actions have effects

177
00:09:37,300 --> 00:09:40,620
on more than one of the subgames of the
compound game.

178
00:09:40,620 --> 00:09:42,270
Okay, now, at first blush, it might seem
that

179
00:09:42,270 --> 00:09:44,760
games of this sort are just plain not
factorable.

180
00:09:44,760 --> 00:09:47,630
However, that's not necessarily true.

181
00:09:47,630 --> 00:09:51,310
Under certain circumstances, even with
interdependent

182
00:09:51,310 --> 00:09:54,730
actions, it's possible to identify factors
and use those factors to

183
00:09:54,730 --> 00:09:58,580
search the game tree more efficiently then
without considering the factors.

184
00:10:00,310 --> 00:10:04,970
Okay here's an example, this is called
Joint Buttons and Lights.

185
00:10:04,970 --> 00:10:07,220
As with the other buttons and lights
variants

186
00:10:07,220 --> 00:10:09,280
that we've seen so far, lights in this
game

187
00:10:09,280 --> 00:10:12,710
are organized in terms of three, however
unlike the

188
00:10:12,710 --> 00:10:15,940
previous variance, buttons are not
associated in specific groups,

189
00:10:15,940 --> 00:10:20,154
instead each button has effects on all
three groups.

190
00:10:20,154 --> 00:10:23,750
Button aaa toggles the first light in each
group.

191
00:10:25,490 --> 00:10:28,220
Button aab toggles the first light in the

192
00:10:28,220 --> 00:10:29,880
first group and the first light in the
second

193
00:10:29,880 --> 00:10:33,300
group and interchanges the values of the
first and

194
00:10:33,300 --> 00:10:36,200
second light in the third group, and so
forth.

195
00:10:36,200 --> 00:10:40,160
So we have all possible combinations of
the actions.

196
00:10:40,160 --> 00:10:41,160
In the preceding games.

197
00:10:44,715 --> 00:10:47,940
Although all of the buttons in this game
affect all of the subgames.

198
00:10:47,940 --> 00:10:51,780
And the game is still factorable with just
three actions per factor.

199
00:10:51,780 --> 00:10:55,350
The reason is that there's one button in
the compound

200
00:10:55,350 --> 00:10:58,910
game for each combination of actions in
the other two subgames.

201
00:10:58,910 --> 00:11:03,080
Thus the game trees for each subgame can
be searched independently of each other

202
00:11:03,080 --> 00:11:08,490
and the actions chosen can be reassembled
into overall actions of the compound game.

203
00:11:08,490 --> 00:11:09,750
For example, if action a is

204
00:11:09,750 --> 00:11:13,160
chosen in the first and second subgames,
and action c is chosen in

205
00:11:13,160 --> 00:11:18,190
the third subgame, then action aac can be
performed in the compound game.

206
00:11:20,975 --> 00:11:23,930
Okay, now recognizing when this can be
done is not easy.

207
00:11:23,930 --> 00:11:24,875
But it is doable.

208
00:11:24,875 --> 00:11:28,630
There are just basically two steps, and
the first step the

209
00:11:28,630 --> 00:11:33,160
player groups actions into equivalence
classes of actions for each subgame.

210
00:11:33,160 --> 00:11:35,920
And in the second step, the player
determines whether these

211
00:11:35,920 --> 00:11:39,590
equivalence classes satisfy what is called
the lossless join condition.

212
00:11:40,920 --> 00:11:44,380
Finding equivalence classes is done by
looking at the propnet for each subgame.

213
00:11:44,380 --> 00:11:46,170
Two actions are equivalent if they are
used

214
00:11:46,170 --> 00:11:48,360
in the same way in the propnet for that
subgame.

215
00:11:48,360 --> 00:11:51,470
For example, if each action is input to an
and gate

216
00:11:51,470 --> 00:11:55,290
with a same other input, and if the
outputs are connected by

217
00:11:55,290 --> 00:11:57,190
an or gate, then the effects of one action
can not be

218
00:11:57,190 --> 00:11:59,790
distinguished from the effects of the
other action, hence they are equivalent.

219
00:12:01,270 --> 00:12:05,630
The process of finding equivalence classes
is repeated for each potential subgame.

220
00:12:05,630 --> 00:12:08,370
In general, the equivalence classes for
each subgame will be different.

221
00:12:08,370 --> 00:12:09,930
In fact, as we'll see, in order for

222
00:12:09,930 --> 00:12:11,770
the game to be factorable, they must be
different.

223
00:12:13,830 --> 00:12:17,860
Once a player has equivalence classes for
each potential subgame, and

224
00:12:17,860 --> 00:12:21,960
then checks whether those equivalence
classes are independent of each other.

225
00:12:21,960 --> 00:12:22,770
Criterion is simple.

226
00:12:22,770 --> 00:12:25,985
Each equivalence class in one potential
subgame must have a

227
00:12:25,985 --> 00:12:28,090
non-empty intersection with each
equivalence

228
00:12:28,090 --> 00:12:30,280
class of every other potential subgame.

229
00:12:30,280 --> 00:12:31,870
If this is true, then these

230
00:12:31,870 --> 00:12:35,160
partitions pass the so-called lossless
join criterion.

231
00:12:35,160 --> 00:12:36,250
More details in the notes.

232
00:12:36,250 --> 00:12:36,750
If

233
00:12:38,370 --> 00:12:43,340
a player finds equivalence classes for
various two for potential subgames.

234
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And they pass this lossless join tests.

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And the player can factor the games into
subgames.

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In order to benefit from the factoring,
the player must modify each propnet so

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that the individual actions are replaced
with

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the equivalence classes of which they are
members.

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This cuts down on the number of possible

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actions to consider, otherwise the
branching factor of the

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00:13:00,120 --> 00:13:01,580
game trees for the subgames will be just
as

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large as the branching factor for the
overall game.

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Once this is done, the player can then
search the game trees

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for the different subgames to select
actions to perform in each subgame.

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They can then find an action in the

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00:13:12,790 --> 00:13:16,090
compound game for the particular
combination of subgame actions.

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This is always doable provided that the

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partitioning of action satisfies that
lossless joins property.

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00:13:21,890 --> 00:13:23,890
the player then performs any action in the

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00:13:23,890 --> 00:13:26,350
intersection of the equivalence classes
chosen in this process.

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Okay, again, check out the notes for more

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details on, on factoring games with
interdependent actions.

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Finally, let's look at one more case,
namely, conditional factoring.

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That is factoring in cases where a game is

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not factorable to start out, but over time
becomes factorable.

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Here's an example.
It's called Linked Tic Tac Toe.

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Two games of Tic Tac Toe connect,
connected by

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a single square that, that connects the
two subgames.

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00:13:55,270 --> 00:13:59,950
The goal of the game as a whole is for the
player to get two lines a row, column, or

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diagonal of that player's mark, with at
least two of

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00:14:02,460 --> 00:14:06,570
the marks residing in a specific Tic Tac
Toe domain.

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Diagonals through the middle, don't count,
diagonals don't count.

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00:14:10,660 --> 00:14:15,980
When in each turn the player in control
can place two marks either one in each, in

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00:14:15,980 --> 00:14:20,350
each distinct Tic Tac Toe domain, or one
mark in the Tic Tac Toe domain, and one

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00:14:20,350 --> 00:14:21,010
in the center square.

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00:14:22,370 --> 00:14:24,799
Now suppose the state of the game is as
shown here.

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Once it is not possible like this to
achieve

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00:14:28,460 --> 00:14:31,740
a row utilizing the center square, the
only possible

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00:14:31,740 --> 00:14:33,632
solution lie in the domains of the Tic Tac

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00:14:33,632 --> 00:14:35,890
Toe games that are conjoined by the center
square.

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00:14:35,890 --> 00:14:39,070
The states of these two Tic Tac Toe games
can be considered

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independently to find the remaining
optimal

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00:14:41,960 --> 00:14:42,990
moves for the duration of the game.

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00:14:44,020 --> 00:14:45,398
Only the game trees for each

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00:14:45,398 --> 00:14:47,840
Tic Tac Toe game, modular the center
square, need

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00:14:47,840 --> 00:14:51,210
to be searched to determine the remaining
optimal moves.

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00:14:51,210 --> 00:14:52,890
Given the current state, the game can best

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be factored into two independent subgames
at this point.

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00:14:56,040 --> 00:14:57,780
However, for the initial state, this game
cannot

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00:14:57,780 --> 00:15:00,470
be factored into sub, independent games
because the shared

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00:15:00,470 --> 00:15:02,740
middle square intertwines the two domains
as it's

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relevant to the satisfaction goals in the,
both subgames.

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00:15:05,645 --> 00:15:10,570
While this game is not factorable, in
general, it is factorable

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00:15:10,570 --> 00:15:13,460
contingent upon entering a state in which
no

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00:15:13,460 --> 00:15:16,430
row through the middle is possible for
either player.

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00:15:16,430 --> 00:15:20,500
So the game is s-, said to be contingently
factorable since it

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00:15:20,500 --> 00:15:23,730
reduces to independent simultaneous
subgames, given

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00:15:23,730 --> 00:15:25,470
that it enters in this certain state.

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00:15:25,470 --> 00:15:26,799
It's contingent upon this state.

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00:15:27,830 --> 00:15:31,980
The raw computational benefit acquired
from recognizing contingent factors like

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00:15:31,980 --> 00:15:35,620
these is less than that of recognizing
initial factors because

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00:15:35,620 --> 00:15:37,528
it requires that the game be in a specific
state.

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00:15:37,528 --> 00:15:41,410
However, the re-, relative computational
savings are still

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00:15:41,410 --> 00:15:44,860
the same because the number of accessible
fringe nodes

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00:15:44,860 --> 00:15:48,440
reduces to the sum of the remaining
accessible fringe

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00:15:48,440 --> 00:15:51,480
nodes of the individual games rather than
their product.

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00:15:53,860 --> 00:15:55,110
Okay, at this point in time, there

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00:15:55,110 --> 00:15:58,510
are no established techniques for
discovering and exploiting

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00:15:58,510 --> 00:16:02,560
cases of contingent, conditional
independence, although there is

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00:16:02,560 --> 00:16:04,950
some work that is being done right now.

301
00:16:04,950 --> 00:16:06,840
However, it seems likely that some of
these techniques

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00:16:06,840 --> 00:16:09,060
that we have already discussed can be
further adapted

303
00:16:09,060 --> 00:16:11,250
or these new techniques that are being
discussed may

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00:16:11,250 --> 00:16:14,840
turn out to have power in finding
conditional factors.

305
00:16:17,270 --> 00:16:20,180
Okay, game factoring is complicated, as
you

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00:16:20,180 --> 00:16:23,860
probably have guessed but it is really
worthwhile.

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00:16:23,860 --> 00:16:26,430
You can significantly decrease the cost of
game playing.

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00:16:26,430 --> 00:16:31,480
Or not the only source of propnet based
game analysis that is worthwhile.

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00:16:31,480 --> 00:16:35,270
For example, propnets can be used to find
bottlenecks.

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00:16:35,270 --> 00:16:38,250
That is, games structured in a series
where the

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00:16:38,250 --> 00:16:41,400
player must win the first game before
playing subsequent games.

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00:16:43,912 --> 00:16:46,550
Propnets can be used to eliminate dead
states, that

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00:16:46,550 --> 00:16:48,930
is states that cannot lead to an
acceptable solution.

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00:16:48,930 --> 00:16:51,830
And which can, therefore, be pruned from
further consideration.

315
00:16:53,648 --> 00:16:55,560
And propnets can also be used to detect

316
00:16:55,560 --> 00:16:58,450
certain kinds of monotonicities, that is,
cases where

317
00:16:58,450 --> 00:17:02,050
higher goal values on non-terminal states
correlate with

318
00:17:02,050 --> 00:17:05,730
progress toward terminal states with
acceptable goal values.

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00:17:05,730 --> 00:17:07,700
And there's a lot more research to be done
in this area.