Team Reasoning
Natalie Gold, King’s College London
Formal Ethics,
Bayreuth, July 2015
I: INTER-PERSONAL TEAM
REASONING
TWO PUZZLES OF GAME THEORY:
Prisoner’s Dilemma and Hi-Lo
Two Puzzles of Game Theory I
The Prisoner’s Dilemma:
P1 cooperate
defect
P2
cooperate defect
3, 3
1, 4
4, 1
2, 2
Two Puzzles of Game Theory I
The Prisoner’s Dilemma:
P1 cooperate
defect
a>b>c>d
b > (a + d)/2
P2
cooperate defect
b, b
d, a
a, d
c, c
Two Puzzles of Game Theory II
Hi-Lo:
P2
P1
high
low
high
2, 2
0, 0
low
0, 0
1, 1
Two Puzzles of Game Theory II
Hi-Lo:
P2
P1
a>b
high
low
high
a, a
0, 0
low
0, 0
b, b
SCHEMATA OF PRACTICAL
REASONING:
Framework for representing
modes of reasoning
Schemata of Reasoning
(1) There are no English mountains over 1000 metres high.
(2) Snowdon is a mountain which is 1085 metres high.
______________________________________
Snowdon is not in England.
Schema 1: Individual Rationality
P1
left
O1
(1)
(2)
(3)
(4)
right
O2
O1  O2
I must choose either left or right.
If I choose left, the outcome will be O1.
If I choose right, the outcome will be O2.
I want to achieve O1 more than I want to
achieve O2.
__________________________________________________________________________
I should choose left
Individually Instrumental Reasoning for P1
in Hi-Lo
(1) I am P1 in Hi-Lo.
(2) The probability that P2 will choose high is 0.5.
_______________________________________
I should choose high.
Collective Choice
P1
left
right
-----P2 -------------------------- P2----left
right
left
right
O1
O2
O4
O3
O1  O2 , O3 , O4
Schema 2: Collective Rationality
(1) We must choose one of (left, left), (left, right), (right, left)
or
(right, right).
(2) If we choose (left, left) the outcome will be O1.
(3) If we choose (left, right) the outcome will be O2.
(4) If we choose (right, left) the outcome will be O3.
(5) If we choose (right, right) the outcome will be O4.
(6) We want to achieve O1 more than we to achieve O2, O3 or
O4.
_______________________________________
We should choose (left, left).
Formalising Team Reasoning: Definitions
• A set S of individuals.
• Each individual has a set of alternative actions, from
which she must choose one, A.
• A profile of actions assigns to each member of S one
element of her set of alternative actions.
• For each profile, there is an outcome.
• A payoff function is a function which assigns a
numerical value to every outcome.
Group Identification
Now consider:
any individual i,
and any set of individuals T,
such that i is a member of T
and T is a weak subset of S.
We will say that i identifies with T if i conceives of
T as a unit of agency, acting as a single entity in
pursuit of some single objective.
Common Knowledge
A proposition p is common knowledge in a set of
individuals T if:
(i) p is true;
(ii) for all individuals i in T, i knows p;
(iii) for all individuals i and j in T, i knows that j
knows p;
(iv) for all individuals i, j, and k in T, i knows that j
knows that k knows that p;
and so on.
Schema 3: Simple Team Reasoning
(from a group viewpoint)
(1) We are the members of S.
(2) Each of us identifies with S.
(3) Each of us wants the value of U to be
maximised.
(4) A uniquely maximises U.
____________________________________
Each of us should choose her component of A.
Schema 4: Simple Team Reasoning
(from an individual viewpoint)
(1) I am a member of S.
(2) It is common knowledge in S that each member
of S identifies with S.
(3) It is common knowledge in S that each member
of S wants the value of U to be maximised.
(4) It is common knowledge in S that A uniquely
maximises U.
____________________________________
I should choose my component of A.
TEAM REASONING COMPARED
WITH
INDIVIDUAL REASONING:
Team Reasoning Solves the Puzzles that
Individual Reasoning Cannot
Team Reasoning solves Hi-Lo
• Assume payoff function U represents the welfare of the
players in the group.
• The players’ payoffs are equal, so make the values of U
equal to the players’ common payoffs.
• Then (high, high) is the profile that uniquely maximises
U.
• Each player can use Schema 4 to reach the conclusion
that she should choose high.
Team Reasoning solves the Prisoner’s Dilemma
• Assume that U treats the players symmetrically, denote
payoff to (C, C) as uC, payoff to (D,D) as uD, payoff to
(D,C) and (C,D) as uF (for ‘free riding’).
• Assume that U is increasing in individual payoffs, which
implies uC > uD.
• Given the condition b > (a + d)/2, it is natural also to
assume uC > uF.
• Then (C,C) is the profile that uniquely maximises U.
• Each player can use Schema 4 to reach the conclusion
that she should choose cooperate.
Is Team Reasoning Necessary?
Team reasoning vs. benefaction
A prisoner’s dilemma with transformed payoffs:
Let a = 10, b = 8, c = 6 and d = 0
Let payoff function for the group {P1, P2} be the
average of the payoffs for the two
P2
cooperate defect
P1 cooperate
b, b
d, a
defect
a, d
c, c
Is Team Reasoning Necessary?
Team reasoning vs. benefaction
A prisoner’s dilemma with transformed payoffs:
Let a = 10, b = 8, c = 6 and d = 0
Let payoff function for the group {P1, P2} be the
average of the payoffs for the two
P2
cooperate defect
P1 cooperate
8, 8
0, 10
defect
10, 0 6, 6
Is Team Reasoning Necessary?
Team reasoning vs. benefaction
A prisoner’s dilemma with transformed payoffs:
Let a = 10, b = 8, c = 6 and d = 0
Let payoff function for the group {P1, P2} be the
average of the payoffs for the two
P2
cooperate defect
P1 cooperate
8, 8
5, 5
defect
5, 5
6, 6
THEORIES OF TEAM AGENCY:
Why think as a team?
How are Teams Formed?
• Rationality/ morality
• Framing
• Mutual commitment
• Non-rational assurance
Team Reasoning When Not All Group
Members Group Identify
• Let T be any subset of S (the team who identify)
• Each member of the team T identifies with S
• Each member of T wants U to be maximised
• U represents what people want as members of S
Team Agency Required by
Rationality/ Morality
Morality Requires Team Agency
• Hodgson (Rule Utilitarianism)
• Regan (Cooperative Utilitarianism)
‘what each agent ought to do is to co-operate, with
whoever else is co-operating, in the production of the
best consequences possible given the behaviour of
non-co-operators’ (1980, p. 124)
Schema 5: Restricted Team Reasoning
(1) I am a member of T.
(2) It is common knowledge in T that each member
of T identifies with S.
(3) It is common knowledge in T that each member
of T wants the value of U to be maximised.
(4) It is common knowledge in T that AT uniquely
maximises U, given the actions of non-members
of T.
____________________________________
I should choose my component of AT.
Rationality/ Morality Requires Team
Agency
• Anderson (principles of self-identification)
– Rationality: identity prior to rational choice (2001, p.
32).
– Morality: requires us to transcend our various
identities and harmonise their demands, by
identifying with a community that comprehends them
all – the Kantian Kingdom of Ends
• Hurley (specify agent-neutral goals)
‘survey the units of agency that are possible in the
circumstances at hand and ask what the unit of
agency, among those possible, should be’; and‘ask
ourselves how we can contribute to the realization
of the best unit possible in the circumstances’ (1989,
pp. 136-159)
Team Agency as the Result
of Framing
Team Membership and Framing
Bacharach (2005)
• A frame is the set of concepts a player uses when
thinking about her situation.
• In order to team reason, a player must have the
concept ‘we’ in her frame.
• Frames can be primed, or induced
• Hi-Lo and the PD both have a property called strong
interdependence
Framing and the Prisoner’s Dilemma
• Bacharach (2005)
‘In a Prisoner’s Dilemma, players might see only, or most
powerfully, the feature of common interest and reciprocal
dependence which lie in the payoffs on the main diagonal.
But they might see the problem in other ways. For
example, someone might be struck by the thought that
her coplayer is in a position to double-cross her by playing
[defect] in the expectation that she will play [cooperate].
This perceived feature might inhibit group identification.’
(2005, Chapter 2, Section 4.2)
T-Conditional Common Knowledge
A proposition p is T-conditional common
knowledge in a set of individuals T if:
(i) p is true;
(ii) for all individuals i in S, if i is a member of T,
then i knows p;
(iii) for all individuals i and j in S, if i is a member
of T, then i knows that if j is a member of T, then
j knows p;
and so on for all individuals i and j in T, i knows
that j knows p;
Schema 6: Circumspect Team Reasoning
(1) I am a member of T.
(2) It is T-conditional common knowledge that each
member of T identifies with S.
(3) It is T-conditional common knowledge that each
member of T wants the value of U to be
maximised.
(4) It is T-conditional common knowledge that A
uniquely maximises U, given the actions of nonmembers of T.
____________________________________
I should choose my component of A.
Circumspection and the Prisoner’s Dilemma
• Set S = {P1, P2}
• Let 0 <  ≤ 1 for each individual be the probability
that the ‘we’ frame comes to mind, in which
case the individual identifies with {P1, P2}
• if this frame does not come to mind, the player
conceives of himself as a unit of agency and thus,
using best-reply reasoning, chooses the dominant
strategy defect
• which protocol maximises U, given the value of
?
Circumspection and the PD cont.:
Optimal profile when in the ‘We’ frame
Two cases to consider:
1) If uF  uD, then (cooperate, cooperate) is the
U-maximising protocol for all possible values of

2) If uD > uF, which protocol maximises U depends
on the value of 
- at high values, (cooperate, cooperate) is uniquely
optimal
- at low values, (defect, defect) is uniquely optimal
Framing is a Psychological Mechanism
• If uF  uD or the value of  is high enough to
make (cooperate, cooperate) the uniquely optimal
protocol, we have a model in which players of the
Prisoner’s Dilemma choose cooperate if the ‘we’
frame comes to mind, and defect otherwise.
• For any given individual, if she identifies with S
and wants U to be maximised, it is instrumentally
rational for her to act as a member of T.
• The parameter  is interpreted as a property of a
psychological mechanism
Team Agency Produced by
Commitment
Commitment and Team Membership
• Gilbert (1989) ‘plural subject’
- formed by public expression of commitment
• Hollis (1998) ‘most remarkable change in man’
- collective act of commitment in Rousseau
Commitment and Framing Accounts Compared
• Bacharach: cannot choose your frame, but opportunities for
cooperation will tend to prime a we–frame
• Gilbert: create plural subject with the rational intention of arriving
at mutually beneficial solutions to problems of coordination or
cooperation
• Bacharach’s concept of framing allows one person to choose an
action with the intention of affecting someone else’s frame.
• ‘Would you like to dance?’
Gilbert: the first stage in a process which may lead to a common
understanding that the two people are a plural subject in relation
to a dance.
Bacharach: part of a process by which two people influence one
another’s frames.
• Schema 4 is compatible with Gilbert.
Schema 4 and Commitment
• On this interpretation schema 4 is not one of
instrumental reasoning.
• The rationality of acting as a member of a team
derives from the rationality of fulfilling one’s
commitments or intentions.
• There is no problem that S may be different
from T.
Team Agency and Assurance
Assurance and Team Reasoning
• Schemas 4, 5 and 6 yield unconditional conclusions
• Schema 4 has assurance
- only tells the agent to choose her component of the
best profile in situations in which it tells the other
members of S chose theirs
- there is common knowledge in S that everyone
identifies with S
• This does not carry over to Schemas 5 and 6:
- each member of the team T identifies with S
- each member of T wants U to be maximised
- U represents what people want as members of S
Logic of Team Reasoning
• Sugden (2003) presents a logic of team reasoning.
• A ‘logic’ is an internally consistent set of axioms and
inference rules, no agent neutral concepts of ‘validity’
and ‘rationality’.
• Represents team reasoning as a particular inference rule
which, as a matter of empirical fact, many people
endorse.
• Re-interpret Schema 4 as an inference rule.
• But what about assurance?
Cross-Personal Reason to Believe
There is cross-personal reason to believe a
proposition p is in a set of individuals T if:
(i) for all individuals i and j in T, where i  j, i has
reason to believe that j has reason to believe p;
(ii) for all individuals i, j, and k in T, where i  j and j 
k, i has reason to believe that j has reason to believe
that k has reason to believe p;
and so on
Schema 7: Mutually Assured Team Reasoning
(1) I am a member of S.
(2) In S, there is common cross-personal reason to
believe that each member endorses and acts on
mutually assured team reasoning with respect to
S and U.
(3) In S, there is common reason to believe that A
uniquely maximises U.
(4) I endorse mutually assured team reasoning.
____________________________________
I should choose my component of A.
Team reasoning in Groups
Theories of team reasoning differ on several dimensions:
• how to deal with cases in which not every member of the
relevant group can be relied on to identify with the group
•
whether group identification is a product of psychological
framing or conscious commitment
• whether each individual’s engaging in team reasoning is
conditional on assurance that others engage in it too
• if so, whether assurance is generated by common
knowledge of the psychology of framing, by joint
commitment, or by experience.
But team reasoning is just as coherent and valid as the bestreply reasoning of conventional game theory.
I: INTRA-PERSONAL TEAM
REASONING
Decision theory and dynamic choice
“The individual over time is an infinity of
individuals” (Strotz, 1955-6)
Dynamic Choice
• at time t, the decision is made by a distinct
transient agent, the person at time t
• self-control as a problem of diachronic consistency:
– the transient agent in a early period would like to
implement a particular series of action but she knows
that when the transient agents of later periods come to
make their choices, they will not have incentives to
carry out their part of the plan.
Backwards Induction
Your corn is ripe to-day; mine will be so to-morrow. ‘Tis
profitable for us both, that I shou’d labour with you today, and that you shou’d aid me to-morrow. I have no
kindness for you, and know you have as little for me. I
will not, therefore, take any pains upon your account; and
should I labour with you upon my own account, in
expectation of a return, I know I shou’d be disappointed,
and that I shou’d in vain depend upon your gratitude.
Here then I leave you to labour alone: You treat me in the
same manner. The seasons change; and both of us lose
our harvests for want of mutual confidence and security.
(Hume, 1739-40/1978, pp. 520-521)
Reason by backwards induction
• Two transient agents,
P1 and P2
• P1: what will P2 do if I
go to the middle?
• Concludes ‘I Should go
to the middle’
Reason by backwards induction
• Two transient agents,
P1 and P2
• P1: what will P2 do if I
go to the middle?
• Concludes ‘I Should go
to the middle’
Reason by backwards induction
• Two transient agents,
P1 and P2
• P1: “What will P2 do if I
go to the middle?”
• Concludes ‘I Should go
to the middle’
Reason by backwards induction
• Two transient agents,
P1 and P2
• P1: “What will P2 do if I
go to the middle?”
• Concludes: “I should go
to the middle”
Present bias: ‘Special contiguity’
In reflecting on any action, which I am to perform a
twelve-month hence, I always resolve to prefer the
greater good, whether at that time it will be more
contiguous or remote; nor does any difference in that
particular make a difference in my present intentions or
resolutions. ... But on my nearer approach, those
circumstances, which I at first over-look’d, begin to
appear, and have an influence on my conduct and
affections. A new inclination to the present good springs
up, and makes it difficult for me to adhere inflexibly to my
first purpose and resolution.
(Hume,1739-40/1978, p. 536)
Strategic interaction between selves
Strategic interaction between selves
• Saturday night at the movies:
Strategic interaction between selves
• Saturday night at the movies:
– Week 1, a mediocre movie
– Week 2, a good movie
– Week 3, a great movie
– Week 4,
Strategic interaction between selves
• Saturday night at the movies:
– Week 1, a mediocre movie
– Week 2, a good movie
– Week 3, a great movie
– Week 4,
• When do you write the report?
Strategic interaction between selves
• Saturday night at the movies:
– Week 1, a mediocre movie
– Week 2, a good movie
– Week 3, a great movie
– Week 4,
• When do you write the report?
Strategic interaction between selves
• Saturday night at the movies:
– Week 1, a mediocre movie 3
– Week 2, a good movie 5
– Week 3, a great movie 8
– Week 4,
13
• When do you write the report?
Solutions
• Alter opportunities (destroy options, precommitments)
• Alter payoffs/ incentives
• But NOT intentions/plans
Parallel between groups and selves:
Externalities and internalities
Pollution as a Prisoner’s Dilemma/
Externality
• Externality: An agent’s action has costs/ benefits
that are not completely captured by the agent
• There is a cost to refraining from polluting
• There are benefits to everyone when the
individual refrains
• For any individual, the cost may outweigh the
individual benefit
• But everyone prefers the situation where no-one
pollutes to that where everyone pollutes
Giving up Smoking:
a sequential prisoner’s dilemma/
internality
• But transient agent gets enjoyment out of smoking a
cigarette, which she would forgo if she refrained
from smoking (a cost)
• Benefit (reduced risk of cancer) is an externality - not
entirely captured by transient agent
• Transient agent rationally “defects”, i.e. smokes
• Transient agent thinks that it is better, over her
lifetime, to be a non-smoker rather than a smoker, as
that reduces the risk of cancer
Utility:
experiences, preferences, welfare
• Decision utility. The utility as inferred
from decisions. This is revealed
preference.
• Experience utility. The pleasure and pain
experienced at a given moment in time
(instant or total).
A Simple Model of Dynamic Choice
(including intra-personal altruism)
•
•
•
•
One person, three periods
In each period t, the individual’s consumption is xt
For each period t, an experienced utility function vt
For each period t, a lifetime utility function ut
ut = βt, 1 v1(x1) + βt, 2 v2(x2) + βt, 3 v3(x3)
βt, r > 0 for all t and r
Example: Jo’s Examination
• In period 3, Jo takes an examination.
• In each of periods 1 and 2, she can choose between revising for
the examination (work) and relaxing (rest).
• The experienced utility of working is –3, that of resting is 0.
• In period 3, experienced utility is
0 if Jo has rested on both previous days,
5 if she has rested on one day and worked on the other,
10 if she has worked on both.
• In the lifetime utility function, the present experienced utility is
given twice as much weight as that of other periods.
Jo’s Examination cont.
Jo1
Jo2
work rest
work 1, 1, 14 –1, 2, 7
rest
2, –1, 7 0, 0, 0
Note: the interaction between Jo1 and Jo2 is a
sequential Prisoner’s Dilemma
Team Reasoning
“What should we do?”
Team Reasoning
“What should we do?”
Why Intra-Personal Team Reasoning?
(1) Indeterminacy of rational choice theory
-challenge the backwards induction solution
(2) Philosophical arguments about agency
- personal utilitarianism vs. agency
Personal Utilitarianism
• Parfit (1984)
– Personal identity is Relation R – overlapping
chains of strong connectedness
– No rational reason to care about future selves
– But moral obligations to look after future selves
– But personal identity is not what matters
– “If I fail to identify with that earlier self, I am in
some ways thinking of that self as a different
person” (Parfit Reasons and Persons, p.319)
Team Agency
• Korsgaard (1989)
– deliberative standpoint from which you choose
– standpoint of practical reason requires unity of agency
– or we couldn’t pursue projects
• Cf. Frankfurt’s ‘Standpoint’ = how agent’s various
attitudes, and other relevant features of her
psychology, come together to constitute a
coherent, relevant standpoint
• Bratman (2014): timeslice evaluative vs. crosstemporal standpoint
Re-cap
• Team reasoning leads to rational cooperation
by people in groups
• Problems of self-control as intra-personal
dilemmas
• Team reasoning leads to rational formation
and following of intentions
Thanks!