This work draws on the Supertask literature1 in order to better understand the conceptual and physical possibility of an infinite decision puzzle presented by Barrett and Arntzenius (1999, 2002). The first section presents the puzzle and two possible objections documented in the literature. The next section argues that cardinality and tracking considerations play a key role in understanding the puzzle. The work concludes with a discussion about some implications for the decision theory.supertask infinite decision puzzle

Garcia-Fronti, Javier

English

Research Papers in Economics

 
 
 
 
A Short Note on the Infinite Decision Puzzle  
Javier García Fronti 
fronti@econ.uba.ar 
 
July 2008 
 
 
 
 
 
 
This  work  draws  on  the  Supertask  literature
1  in  order  to  better  understand  the 
conceptual and physical possibility of an infinite decision puzzle presented by Barrett 
and Arntzenius (1999, 2002).  The first section presents the puzzle and two possible 
objections documented in the literature. The next section argues that cardinality and 
tracking  considerations  play  a  key  role  in  understanding  the  puzzle.  The  work 
concludes with a discussion about some implications for the decision theory. 
 
 
 
 
 
 
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1 Benacerraf 1962, Black 1951, Laraudogoitia 1996, 1999 and Silagadze 2005. 2 
 
1. The Puzzle 
 
Barrett and Arntzenius (1999, 2002) present an economy populated by two players: a 
Bank and an agent. At time t=0, the first one has an infinite stack of $1 bills numbered 
following the natural numbers (.i.e. 1, 2, 3….). Meanwhile the agent has no wealth.  
 
In this context, the bank offers two possible alternative exchange proposals to the 
agent: 
a.  Receive the bill that is in the top of stack in any exchange, 
or 
b.  receive the top 3 bills and return to the bank that one holding the small serial 
number in any exchange. 
 
Assuming that there is only one exchange between the bank and the agent; a rational 
individual always chooses option (b) netting $2 (bigger than option (a) netting $1). 
Moreover, if the exchange takes place n times the payoff of each option are: 
a. 
1
$1 $
n
k
n

              <               b. 
1
$2 $2
n
k
n

  . 
Hence, as far as n is bounded, a rational agent always prefers option (b). 
 
1.1. An Infinite Problem  
 
On the contrary, if infinite exchanges are allowed, the setup of the problem and so the 
decision faced by the agent changes dramatically. Assuming that the time sequence is 
t = 0,1/2, 3/4, 7/8,…   i.e.
21
/:
2
k
k k k t

    ,  Now  the  bank  and  the  agent  are 
involved in a Supertask
2. 
 
In this setup, each sequential task is an exchange of bills between the bank and the 
agent. Which is the agent‟s best strategy in this infinite setup? The authors note that if 
she chooses option b, she will end up with nothing at t=1: every bill numbered k in 
her hands will be back to the bank at instant t=1-1/2
K. On the contrary, if the agent 
chooses (a), the entire bank stack will end in her hands!  
 
Hence, in the infinite setup, the more profitable strategy is (a).  
 
While in finite time the agent always chooses (b), in an infinite setup it is optimal to 
choose (a). Is it a paradox? There are two possible objections regarding the existence 
of this infinite decision puzzle already mentioned in the recent literature. 
 
                                                 
2  See Laraudogoitia (1999) for detailed presentation. 3 
 
1.2. Two Objections  
 
 
Machina (2000) argues that because the end state (t=1) is not well defined there is no 
conceptual possibility for the proposed supertask. However, the supertask actions are 
taken place for t<1, at t=1 there is no remain actions (Benacerraf 1962). The t-series 
includes all the instants where actions takes place (t<1). The supertask rules for every 
t<1, therefore, the state of the world at t=1 could not be a logical consequence of the 
previous t-series.  
 
Pulier (2000) states that the infinite sum is not well defined, so he argues that the 
proposed supertask is no possible. Nevertheless, Benacerraf (1962) clearly states that 
the properties shared by the terms of a succession could not be valid whenn . 
 
Barrett and Arntzenius (2002) defend their position noting that the puzzle proposed 
has no relation with the summation of the wealth of the agent; on the  contrary it 
relates to the individual bills hold by him. They build a problem where one could 
track the physical trajectory of every bill from the bank to the agent and it return to 
the bank. The trajectories are always defined and can establish the final situation of 
the agent at time t=1. This is a coherent representation of the problem and has a 
mathematical structure. So the puzzle remains valid.  
 
The  argument  presented  by  Barrett  and  Arntzenius  is  an  application  of  the 
Benacerraf‟s Critique (1962). He postulates that the arguments about the end state and 
the  properties  of  the  partial  sum  do  not  demonstrate  the  logical  impossibility  of 
supertasks. He based his argument on two main ideas (Pérez Laraudogoitia, 2004): 
 
(a) The state of the world at an instant immediately after the end of a supertask 
does not necessary follow previous states during the supertask. 
(b) The “finite” properties shared by the terms of a succession could not be 
valid for the series when the number of terms tends to infinite.  
 
Summing  up,  the  two  arguments  against  the  proposed  supertask  by  Machina  and 
Pulier could be refuted using Benacerraf (1962).  
 
2.  Cardinality and tracking considerations 
 
 
Scott and Scott (2005) note that the infinite decision puzzle presented above rest on 
the mathematical concept of equi-cardinality and the physical possibility of tracking 
bills. Equi-cardinality between the bills returned to the bank and the natural numbers; 
and the physical possibility of tracking the bills movement in an Euclidean space 
following classical mechanics. 
 4 
 
2.1. Cardinality 
 
Scott and Scott (2005) postulate that mathematical cardinality and the possibility of 
tracking the bills anytime are crucial in the correct understanding of the problem. 
Assuming the same infinite timing as before, one could interpret option (b) as one in 
where there are two action sequences at any stage:  
 
1.   The agent receives from the bank the top 3 bills at any time t.  Define A
k as the 
set of bills hold by the agent at stage k. 
2.   The agent returns the bill holding the small serial number in any exchange. 
Define B
k as the set of bills hold by the bank at stage k. 
 
There is a bijection between these two sets, i.e. / ( ) 3
KK A B f k k  , so they have 
the same cardinality.  
 
2.2. Physical possibility 
 
However, this is only a necessary condition for two infinite sets to be equal; having 
the same cardinality does not mean “identical”. 
 
Assuming that the previous justification convinces the reader about the conceptual 
possibility  of  the  proposed  supertask,  let‟s  now  analyse  its  potential  physical 
possibility.  
 
Following  Grünbaum  (1970)  and  Pérez  Laraudogoitia  (2004),  a  supertask  is 
kinematically possible if the following two conditions are satisfied: 
  1. All the bills travel at bounded speed 
  2.  :0 1 tt    , the position of all the moving bills approach a defined limit. 
 
Richard  Sainsbury  (1988) suggests  that  the  continuity principle  could  solve  some 
infinite  puzzles  (e.g.  supertasks).  The  principle  states  that  the  trajectories  of  the 
moving bodies are continuous lines. Formally, for every body i: 
    :lim ( ) ( ) ii ut t X u X t
   
where  i X  is the position of i.  
 
Please note that there is no final exchange in our experiment. Regarding to physical 
world assumed here, the bills are bodies in Euclidean space. Moreover, the position is 
a function continuous in time. Formally, the space is a plane (x,y) and the position is 
defined by a function  ( , , ) f x y t continuous in x, y and t. 
 
The second condition for equality in an infinite framework proposed by Scott and 
Scott (2005) is that it must be possible to track each bill movement from A to B (i.e. 
the existence of a physical trajectory). Formally, this is done by noting that every bill 
with serial number “k” will be returned to the bank at 
1
1
2
k t  .  
 5 
 
Hence, at the end of the supertask the two sets are equals (A=B), and if the agent 
chooses option (b) he will end with no bills. 
 
In a different physical setup, tracking could not be possible. For example, in a world 
that follows quantum mechanics laws, it is no possible to track each bill movement. 
One  could  only  talk  about  fall  or  increase  of  the  number  of  bills,  with  no 
individualisation. In this physical world, only cardinality apply so the puzzle remains 
because it is possible to have infinitely bills at t=1.  
 
3.  Implications for the decision theory 
 
Barrett and Arntzenius (2002) describe a similar puzzle where the wealth of the agent 
is finite. They assume that each additional bill has positive but decreasing marginal 
utility, so the total wealth in any case is a finite amount. This example clearly shows 
that the puzzle is not destroy by supporting the idea that there is no infinite wealth in 
the real world, on the contrary, it is based on infinite but identifiable trajectories of 
transaction of goods. On the other hand, the authors accept that the puzzle requires an 
agent confronting an infinite sequence of decisions. 
 
What about the energy needed for moving the bills? In the original version the authors 
assume infinite. However, the also describes an  alternative puzzle where the total 
energy is upper bounded: each bill, in any subsequent transaction is left closer to the 
frontier between the bank and the agent, so the total energy used remains finite. This 
example assumes that there is no minimum energy required for a single transaction. 
(Barrett and Arntzenius 2002) 
 
Conclusion 
 
Barrett and Arntzenius (1999,  2002) proposed a puzzle where one-step rationality 
contradicts the global rationality. Cardinality and tractability considerations give a 
proper account of the puzzle. Moreover, in the context of business decision problems 
(problems that include physical bills), there is no possibility of separate the logical 
proposal from the physical possibility of the supertask (Scott and Scott 2005).  
 
 6 
 
References 
 
Allis, V. and Koetsier, T., 1991, „On Some Paradoxes of the Infinite‟, British Journal 
for the Philosophy of Science, 42, pp. 187-194  
Allis,  V.  and  Koetsier,  T.,  1995,  „On  Some  Paradoxes  of  the  Infinite  II‟,  British 
Journal for the Philosophy of Science, 46, pp. 235-247  
Barrett,  J.  A.  and  Arntzenius,  F.  (1999)  An  Infinite  Decision  Puzzle.  Theory  and 
Decision, 46 (1), 101-103. 
Barrett, J. A. and Arntzenius, F. (2002) Why the Infinite Decision Puzzle Is Puzzling. 
Theory and Decision, 52 (2), 139-147. 
Benacerraf, P. (1962) Tasks, Super-Tasks, and the Modern Eleatics. The Journal of 
Philosophy, 59 (24), 765-784. 
Black, M. (1951) „Achilles and the Tortoise‟. Analysis, 11 (5), 91-101. 
Earman, J., and Norton, J. (1996) „Infinite Pains: The Trouble with Supertasks‟, in 
Benacerraf and His Critics, A. Morton and S. Stich (eds.), Oxford: Blackwell, 
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Grünbaum, A. (1970) „Modern Science and Zeno's Paradoxes of Motion‟, in Salmon 
[1970], pp. 200-250 
Pérez Laraudogoitia, J. (1996) A Beautiful Supertask. Mind, 105 (417), 81-83. 
Pérez  Laraudogoitia,  J.  (2004)  "Supertasks",  The  Stanford  Encyclopedia  of 
Philosophy  (Winter  2004  Edition),  Edward  N.  Zalta (ed.),  URL  = 
http://plato.stanford.edu/archives/win2004/entries/spacetime-supertasks/>.  
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A Short Note on the Infinite Decision Puzzle

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