The paradox
Game theory
Improv as game
Bayesian inference
A greater-good solution
Conclusion
References
The ‘split agent’ problem (or McGonagall paradox)
in improvised theatre
GCCR HDR Symposium 2014
Griffith University
November 17, 2014
Lochlan Morrissey
l.morrissey@griffith.edu.au
School of Languages and Linguistics
Griffith University, Brisbane, Australia
The paradox
Game theory
Improv as game
Bayesian inference
A greater-good solution
(i.) a character never selects her own actions;
Conclusion
References
The paradox
Game theory
Improv as game
Bayesian inference
A greater-good solution
Conclusion
References
(i.) a character never selects her own actions;
(ii.) for a performance to be compelling, each character must be
internally consistent, and therefore must have motivations that
drive her actions (cf. Bruce et al., 2000; Riedl and Young,
2010): the character should, then, be considered cognizant.
The paradox
Game theory
Improv as game
Bayesian inference
A greater-good solution
Conclusion
References
(i.) a character never selects her own actions;
(ii.) for a performance to be compelling, each character must be
internally consistent, and therefore must have motivations that
drive her actions (cf. Bruce et al., 2000; Riedl and Young,
2010): the character should, then, be considered cognizant;
(iii.) the actor, who is also cognizant, has knowledge of both the
narrative and metanarrative elements of the performance, the
latter of which are not available to the character.
The paradox
Game theory
Improv as game
Bayesian inference
A greater-good solution
Conclusion
References
(i.) a character never selects her own actions;
(ii.) for a performance to be compelling, each character must be
internally consistent, and therefore must have motivations that
drive her actions (cf. Bruce et al., 2000; Riedl and Young,
2010): the character should, then, be considered cognizant;
(iii.) the actor, who is also cognizant, has knowledge of both the
narrative and metanarrative elements of the performance, the
latter of which are not available to the character;
∴ the character’s preferences over the outcome of a plot may
differ from those of the actor who selects her actions.
(N.b: Noted before in e.g., Magerko et al. (2009); Fuller and
Magerko (2010); Baumer and Magerko (2009, 2010)).
The paradox
Game theory
Improv as game
Bayesian inference
A greater-good solution
Conclusion
A (most basic) game includes the following elements:
I
a set of players N = {1, . . . , n };
I
a set of actions α; and
I
a mechanism of assigning real-numbered utilities to plays
U : π → Rn . A play is simply a sequence of actions:
π = (a1 , . . . , an ); ai ∈ α.
References
The paradox
Game theory
Improv as game
Bayesian inference
A greater-good solution
A simple example: matching coins
I
Players are i and j.
I
Possible actions are h and t.
I
Possible plays are (h, t ), (h, h ), (t, t ), (t, h );
I
Utility of any particular play is u (h, t ) = (0; 1).
i, j
h
t
h
(1; 0)
(0; 1)
t
(0; 1)
(1; 0)
Conclusion
References
The paradox
Game theory
Improv as game
Bayesian inference
A greater-good solution
Conclusion
Private information
I
i’s private information expressed as a type θi , usually
determines utilities; assigned by player 0.
The coin matching game with private information:
0
I
I
h
θ
i
s1
θ0
i
s2
t
j
h
(1; 0)
j
t
h
(0; 1)
(0; 1)
t
(1; 0)
Note that utilities become expected utilities.
Each player has beliefs regarding her fellow players’ types,
based on her own.
References
The paradox
Game theory
Improv as game
Bayesian inference
A greater-good solution
Conclusion
References
An improvised performance game is:
I
a set of players, partitioned into two cells: actors and roles;
I
a set of allowable actions according to the rules of the game,
or restrictions on the world;
I
a function which assigns utility (based on ad hoc notions of
success) each for actor and role.
The paradox
Game theory
Improv as game
Bayesian inference
A greater-good solution
Conclusion
References
We reframe players as dyadic players: (a, r ).
N = A×R
The following sets of each constituent of the dyadic player are
‘merged’:
I
action sets
I
belief sets
However, it is important to note that type sets remain distinct, and
that a dyadic player’s type is a pair of types such that (θa , θr ). It is
important to preserve distinct types for actor and role, so that their
utilities for any particular play remain distinct.
The paradox
Game theory
Improv as game
Bayesian inference
A greater-good solution
Conclusion
Beliefs
Because the actor does not share a type with her role, she has
beliefs regarding her role.
So I started thinking like, ‘What do I make my wife?’ He
wishes I didn’t have my wife and I’m thinking, ‘Should I not
like my wife or should I be in love with my wife?’ And that’s
going to base how I feel about what he’s saying to me. My
opinion about the woman I’m married to. So I could’ve sided
with him and said, which I almost said; I almost said, ‘I wish I
wasn’t married to her either. She’s a real [expletive].’ But I
decided to take the other stance and going like, ‘Why? What’s
wrong with my wife?’ and then thinking like, ‘I’m completely
finding sort of a game within the scene. I’m going to be the
total outsider from this trio of friends. I’m going to be the guy
whose opinion differs from them, no matter what.’
(Magerko et al., 2009, 122)
References
The paradox
Game theory
Improv as game
Bayesian inference
A greater-good solution
Conclusion
References
Bayes’s equation
A method of quantifying the influence of evidence on probabilistic
reasoning.
Pr(P |R ) × Pr(R )
Pr(R |P ) =
Pr(P )
The paradox
Game theory
Improv as game
Bayesian inference
A greater-good solution
Conclusion
References
Effects on role construal
An actor’s belief of her role’s type, given her own type, is expressed
as
Pr(θr |θa )
A role-as-character in a plot is construed by considering which
actions are more likely, given a perceived set of attributes.
[. . . ] ‘Should I not like my wife or should I be in love
with my wife?’ And that’s going to base how I feel about
what he’s saying to me. My opinion about the woman I’m
married to. [. . . ]
The paradox
Game theory
Improv as game
Bayesian inference
A greater-good solution
Conclusion
References
Effects on interaction with other roles
I
Actors must judge the types of the other actors, and also those
of their roles. Both of these involve signalling.
I
The types of the other roles affects the attributes an actor
ascribes to her own.
I
The types of the other actors dictates the trajectory and actor
perceives as possessing the highest utility.
The paradox
Game theory
Improv as game
Bayesian inference
A greater-good solution
Conclusion
Effects on utility
The selection of an optimal action, then, must account for
1. the role’s preference, given the actor’s beliefs regarding the
role;
2. the actor’s preference, given her beliefs regarding the other
actors’ preferences (where we assume that the actors are
cooperating);
3. the actor’s preference, given her beliefs regarding the other
roles’ preferences.
References
The paradox
Game theory
Improv as game
Bayesian inference
A greater-good solution
Conclusion
References
Macbeth improvised
We assume that i, the actor playing Macbeth believes the
following: the other actor j wants Macbeth (b) to die, as does his
role, Macduff (d). In this version i is cooperative.
i
j
b
d
dies
1
1
−1
1
lives
−1
−1
1
−1
So, while character consistency is paramount, it is clearly in the
‘greater good’ for Macbeth to die.
The paradox
Game theory
Improv as game
Bayesian inference
A greater-good solution
Conclusion
References
Macbeth improvised: The McGonagall case
Now we assume that McGonagall i does not want to die, so he is
not upstaged by his rival actor j. i’s beliefs regarding his fellow
actors are identical in this version of the game.
i
j
b
d
dies
−1
1
−1
1
lives
1
−1
1
−1
There is no obvious best utility for the group. There is, however, a
best utility for McGonagall and his role.
The paradox
Game theory
Improv as game
Bayesian inference
A greater-good solution
Conclusion
References
The solution
I
Consistent, iterative assessment of a fellow actors’ preferences
allows for greater-good moves and a utilitarian rationale for
following the will of the company, and of the audience, even if
this contradicts the internally consistent preferences of the role.
I
When there are more actors, this effect will be amplified, as
each actor will probably prefer Macbeth to die (as it is
presumably preferred by the audience).
I
Such an approach can account for seemingly selfish or ignorant
moves by an actor: contraventions of greater-good moves,
including those that are accidental.
The paradox
Game theory
Improv as game
Bayesian inference
A greater-good solution
Conclusion
References
I
The McGonagall paradox describes that an actor’s preferences
over the outcome of a performance may differ from her role’s
preferences regarding the plot. We assume that character
consistency is paramount.
I
A game theoretic account of improvised theatre views both the
actor and her role as players of the game, linked as dyadic
players.
I
An actor’s beliefs regarding her role’s, her fellow actors’, and
the other roles’ types affect her expected utilities for any given
play.
I
Thus can we account for decisions made by actors which
contradict their roles’ preferences.
The paradox
Game theory
Improv as game
Bayesian inference
A greater-good solution
Conclusion
References
Baumer, A. and Magerko, B. (2009). Narrative development in improvisational
theatre. In Iurgel, I. A., Zagalo, N., and Petta, P., editors, Interactive
Storytelling, pages 140–151. Springer, Berlin.
Baumer, A. and Magerko, B. (2010). An analysis of narrative moves in
improvisational theatre. In Aylett, R., Lim, M., Louchart, S., and Petta, P.,
editors, Interactive Storytelling, volume 6432 of Lecture Notes in Computer
Science, pages 165–175, Berlin. Springer.
Bruce, A., Knight, J., Listopad, S., Magerko, B., and Nourbakhsh, I. (2000).
Robot improv: Using drama to create believable agents. In Proceedings of
the IEEE International Conference on Robotics and Automation (ICRA ’00),
volume 4, pages 4002–4008.
Fuller, D. and Magerko, B. (2010). Shared mental models in improvisational
performance. In Proceedings of the Intelligent Narrative Technologies III
Workshop, Monterey, CA. ACM.
Magerko, B., Manzoul, W., Riedl, M., Baumer, A., Fuller, D., Luther, K., and
Pearce, C. (2009). An empirical study of cognition and theatrical
improvisation. In Proceedings of the seventh ACM conference on Creativity
and cognition, Berkeley, CA, pages 117–126.
Riedl, M. O. and Young, R. M. (2010). Narrative planning: Balancing plot and
character. Journal of Artificial Intelligence Research, 39(1):217–268.