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\begin{center} {\Large {\bf notes on the Belief Game Between God and Man  }  } \\

 

 THIS IS PRELIMINARY.  



 \bigskip 

13 November  2005 \\


\bigskip

Eric Rasmusen \\



\bigskip

{\it Abstract}
  \end{center}

\begin{small}
  Game theory can be helpful in theology. It provides a new angle on the meaning of free will. 

\bigskip

\end{small}

\noindent {\small \hspace*{20pt}  Dan R. and Catherine M. Dalton Professor, Department  of Business Economics and Public Policy, Kelley School of Business, Indiana University,  BU 456, 1309 E. 10th Street, Bloomington, Indiana, 47405-1701. Office: (812) 855-9219. Fax: 812-855-3354. Erasmuse@indiana.edu. http://rasmusen.org. Copies of this paper can be found at\\
  http://rasmusen.org/papers/theology-rasmusen.pdf.   }

 {\small I thank  xxx. }

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\noindent
  {\bf 1. Introduction}
 

Brams has a game. He didn't do it right.  His game has believe/don't believe. Is that voluntary? 

Also, what he really means is ``believe only if there is evidence'', which is irrational. 

 Also, as he says, the choice is really theist, atheist, agnostic. 

2. A second version is *really* believe/don't believe.  Then you go with your priors. This is perhaps what God wants. 

3. A third version is  Investigate/Don't Investigate.  If you aren't going to investigate, God will not provide the evidence. 

 


   \bigskip 

Table 1 is Figure 2.1 from Brams. 

\begin{center} {\bf Table 1: The  Brams Revelation Game } 

 \begin{tabular}{lllccc} &          &
&\multicolumn{3}{c}{\bf Man}\\
 &      &       & Believe   &    &  Don't Believe \\ 
&  &  Reveal       &    3,4   &    & 1,1 \\
 & {\bf God}    &          &    &     &   \\
 & & Conceal    & 4,2      &  &   2,3 \\
 &      &          &    &     &   \\
  \multicolumn{6}{l}{\it Payoffs to: (God, Man). 4 is best, 1 is worst.}\\
 \end{tabular}\\
 \end{center}

 


Craig Duncan distinguishes between two kinds of agnosticism. One kind of agnostic believes that the existence or nonexistence of God cannot be known, neither now nor in the future. Another kind believes  that the existence or nonexistence of God cannot be known on presence evidence, but does not rule out the possibility that new evidence will arise.   

Whether one can choose to believe is a question taht comes up in Pascal's Wager too, which has a philosophical literature. 

Table 1 has two flaws. One is that a person cannot choose between belief and disbelief if he is rational. All he can do is decide what actions to take based on his beliefs. We say in daily life that somebody decides to believe in something, but we just mean that they understand the facts and theory in a certain way. Once they do, they have no choice but to believe. Or, perhaps they can purposely choose to believe  what they currently believe is false, by crippling their minds somehow. We will return to that with the idea of the Belief Pill below. That is not in game theory, though. 

 The second flaw is that it is a very strange payoff function to have someone believe only if he observes  evidence, even if rationally he should believe even without  new evidence.  If someone has a good theoretical reason to believe that God exists, and he knows that God wants to remain concealed, he should not expect to see any evidence, and he should believe anyway. 

 No:  a better decision for the Man is whether to Obey or Sin. His payoffs from these things depend on whether God exists or not. 


\begin{center} {\bf Table 2:   The  Hard Decision Payoff Matrix  } 

 \begin{tabular}{lllccc} &          &
&\multicolumn{3}{c}{\bf Man}\\
 &      &       & Obey   &    & Sin  \\ 
&  &   Exists  &    4   &    &   1\\
 & {\bf God}    &          &    &     &   \\
 & & Fictional    &  2       &  &  3 \\
 &      &          &    &     &   \\
  \multicolumn{6}{l}{\it Payoff to    Man. 4 is best, 1 is worst. }\\
 \end{tabular}\\

 \end{center}

In Table 2, I am not assuming that Christian salvation comes from obedience to God. That is Pelagianism,  a heresy. Rather, I am assuming that the punishment of someone who has not achieved salvation depends on the degree of their sin. There are circles of Hell, as in Dante, and although someone whom God  has not granted salvation goes to Hell, if he has led a typical life he will not suffer as much as someone who steals, murders, and blasphemes. 

 I am also putting aside sins such as  excessive drinking or habitual lying that are not just displeasing to God, but stupid even for an atheist. ``Obey'' refers only to laws whose validity depends on God's existence (and, of course, His caring about them). 


 This allows us to escape Pascal's Wager, by an outlet even Pascal must grant us. 

Whether the man will obey or sin depends on his beliefs about God. In the example above, he will choose Obey if puts the probability of God's existence at greater than xxx.  The threshold depends crucially on the size of the payoffs now. If  we replace 1 with -100, more appropriate to Pascal's Wager, then the threshold drops to xxx. 

  

 We now return to the  Brams  Revelation Game.  Table 3 represents a bayesian game, one with two possible states of the world. In State A, God exists and is playing a game with the Man. In State B, God does not exist, and only the Man has a choice to make. The Man does not know which Game he is playing. He observes either Strong Evidence or Weak Evidence before making his choice.  (We could make this more general by allowing all degrees of evidence, and putting a probability that even Strong Evidence might appear when God does not exist, purely by chance.) 

\begin{center} {\bf Table 3:   The  Hard Decision Payoff Matrix  } 

 \begin{tabular}{lllccc} &          &
&\multicolumn{3}{c}{\bf Man}\\
 &      &       & Obey   &    & Sin  \\ 
&  &   Strong Evidence       &    3,4  &    &  1,1 \\
 & {\bf God}    &          &    &     &   \\
 & & Weak Evidence     & 4,4        &  &   2,1  \\
 &      &          &    &     &   \\
  \multicolumn{6}{l}{\it Expected Payoffs to: (God, Man). 4 is best, 1 is worst. }\\
 \end{tabular}\\
 State A: God Exists\\

 \begin{tabular}{lllccc} &          &
&\multicolumn{3}{c}{\bf Man}\\
 &      &       & Obey   &    & Sin  \\ 
 &       &          &    &     &   \\
 & & Weak Evidence     & 2       &  &   3 \\
 &      &          &    &     &   \\
  \multicolumn{6}{l}{\it Expected Payoffs to: (Man). 4 is best, 1 is worst. }\\
 \end{tabular}\\
State  B: God Does Not Exist\\

 \end{center}

Suppose the Man has a prior probability of $p$ on God's existence. If he observes  Strong Evidence, he will choose Obey, for a payoff of 4 instead of 1. What he does if he observes Weak Evidence can depend on what he thinks about God's decision. If he thinks God would choose Strong Evidence, then if he observes Weak Evidence, he can conclude God does not exist, and so he will choose Sin (for a payoff of 3) instead of Obey (for a payoff of 2). 

If the Man thinks God, if He exists, would choose  Weak Evidence, then observing Weak Evidence provides no useful information. The Man's expected  payoff from Obey is $p4 + (1-p)2$. His expected payoff from Sin is p*1 + (1-p)*3. These are equal if $4p+2-2p = p +3-3p$, or $2p+2=3-2p$, or $p^*=.25$. For higher $p$, he will believe; for lower $p$, he will not. 




Thus, the outcome depends on the prior probability.  If this is given by God, then it would be a way to explain how God can choose whether some people obey  and some do not---  but indirectly. Whether the people retain ` free will in such a situation  depends on how one defines ``free will''. 


 In State A, God exists. Which will He choose?  Suppose the Man's prior is $p=.5$, so he will choose Obey even if he observes Weak Evidence.   Given the specified payoffs, God's payoff will be 4 from Weak Evidence and 3 from Strong Evidence, so He will choose Weak Evidence. 

If, on the other hand, $p<.25$, the Man will choose Sin if he observes Weak Evidence. Then God's payoff will be 2 from Weak Evidence and 3 from Strong Evidence, and God will choose Strong Evidence. 

Putting these together: God will choose Weak Evidence if and only if the Man will believe anyway. 

 What, now if there is a large group of Men, all with different priors? If God must choose one action for all of them, it will not depend just on the average prior. Suppose the cumulative distribution of priors is $F(p)$ and the total mass of Men is $M$. God's payoff from Strong Evidence  would be simply $3M$, since all would obey. His payoff from Weak Evidence would be $4F(p)M + 2 (1-F(p))M$, which might be higher or might be lower, depending on the $F(p)$ function. 

Yet another alternative is that not all Men have the above payoff function. Some would choose Sin even if there were Strong Evidence. Then God might choose Weak Evidence to avoid damning those Men even more, for payoffs of 1 to God. 

Personally, I do not think  this analysis is very useful, since I do not think that God's payoff from Obey is higher with Weak Evidence than with Strong Evidence. Some people might believe that, however, so I write this for them.  My own belief is that God has other, mysterious, reasons for choosing Weak Evidence, reasons having more to do with the  constraints of the supernatural world  than with maximizing the payoff from saving souls. 


\bigskip

\noindent
{\bf Pascal's Wager and the Myth of Er Argument}

What Pascal's Wager does is replace the payoff of 4 from believing in a true God with a very large payoff, big enough to change the critical prior belief level from $.25$ to some very small size such as .000001. 

There is an alternative to Pascal's Wager, an idea which must be known but which does not have a name. It is to change the  payoff of 3 from Sin in the absence of God to 1.5--- that is, to a level worse than Obey, though better than the payoff of 1 from Sin when God does exist. The interpretation of that is that to obey the laws of a supposed God will make the Man happier than to disobey them, even if God does not exist. We might call this, ``The Myth of Er Argument'', since it is reminds us of the Myth of Er  at the end of  Plato's Republic. Socrates has spent the entire dialog trying to resolve the question posed by the Ring of Gyges: if a man can do evil without fear of punishment (because he has a ring of invisibility, for example), why not do evil? He has been unable to come up with a solid answer, and so he ends the dialog by telling a story in which evil is punished in the afterlife, by a bad reincarnation.  The idea seems to be that a man will be better off if he believes that myth than if he does not, because doing good is better for him than doing evil and without the myth he cannot be convinced to do good. 

 The Myth of Er Argument is, of course, no more consistent with orthodox Christianity than Pascal's Wager, and rather more distasteful.  It says, `` Believe in God, because even if He does not exist, you will be happier believing in Him, for purely earthly reasons.'' 

  
 
 \bigskip

Now let us return to the question of whether one can choose to believe. In Table 3, the decision was made despite lack of complete confidence by the  Man.  Instead, his decision could be whether to take a ``Belief Pill'' or  an ``Unbelief Pill''. These pills would give him complete confidence in his belief. His value for $p$ would go to either 0 or 1. 



\begin{center} {\bf Table 4:   The Belief Pill Game } 

 \begin{tabular}{lllccc} &          &
&\multicolumn{3}{c}{\bf Man}\\
 &      &       & Take Belief Pill    &    &  Don't Take \\ 
&  &  Reveal       &    3,2   &    &  -1,3 \\
 & {\bf God}    &          &    &     &   \\
 & & Conceal    & -1,1       &  &   0,0 \\
 &      &          &    &     &   \\
  \multicolumn{6}{l}{\it Payoffs to: (God, Man). 4 is best, 1 is worst. }\\
 \end{tabular}\\

 \end{center}

In this simple setting, the Belief Pill makes no difference. The Man can obey either with or without certainty. If he is subject to weakness of will, akrasia, maybe the Belief Pill will help--- a sort  of ``Dutch courage'' (as a swig of whisky before a fight has been called). Or, the payoffs may be affected by the strength of belief, and a person may be happier if he can suppress his doubts and stop thinking about a subject.  On the other hand, if new information comes up--- if, say, God does decide to reveal Himself after all, or conclusive evidence turns up that the Bible was forged--- then having removed one's ability to change one's mind is a disadvantage. 

\bigskip

Finally, let us look at a rather different game, involving the question as to whether to inquire further into the evidence for God's existence. This is another way to alter one's beliefs, a rational way. This will be a coordination game, with two  


\begin{center} {\bf Table 5a:   The Seeker's Game: God Wishes to Save} 

 \begin{tabular}{lllccc} &          &
&\multicolumn{3}{c}{\bf Man}\\
 &      &          &  Inquire   &    &  Ignore \\ 
&  &  Reveal       &    4,3   &       &  2 ,2 \\
 & {\bf God}    &          &    &     &   \\
 & & Conceal    &       1,1       &  &    3,2 \\
 &      &          &    &     &   \\
  \multicolumn{6}{l}{\it Payoffs to: (God, Man). 4 is best, 1 is worst.}\\
 \end{tabular}\\
 \end{center}

 
\begin{center} {\bf Table 5b:   The Seeker's Game: God Wishes to Save} 

 \begin{tabular}{lllccc} &          &
&\multicolumn{3}{c}{\bf Man}\\
 &      &          &  Inquire   &    &  Ignore \\ 
&  &  Reveal         &    1,3   &      &  3, 2 \\
 & {\bf God}    &          &    &     &   \\
 & & Conceal    &       4,1       &  &    2,2 \\
 &      &          &    &     &   \\
  \multicolumn{6}{l}{\it Payoffs to: (God, Man). 4 is best, 1 is worst.}\\
 \end{tabular}\\
 \end{center}

 I am wondering whether God's choice of expectations could be  a way for him to control the outcome.  Maybe not. 





\begin{center}
 {\bf References} 
\end{center}


Aumann, Robert (2003) ``Risk Aversion in the Talmud,'' {\it  Economic Theory},  21:  233-
239 (2003).

Aumann, Robert \& Michael Maschler (1985) ``Game-Theoretic Analysis of a Bankruptcy Problem from the
Talmud,'' {\it  Journal of Economic Theory,} 36: 195-213 (1985).
 
 
  Brams, Steven J.  (1983)  {\it Superior Beings: If They Exist, How Would We Know?}
New York: Springer-Verlag  (1983). 


 
 
 Brams, Steven J.  (1980)
``Mathematics and Theology: Game-Theoretic Implications of God's
Omniscience,''  {\it
Mathematics Magazine},    53(5):  277-282 (November 1980). 

 Brams, Steven J. 
``Belief in God: A Game-Theoretic Paradox,'' {\it International Journal for
Philosophy
of Religion,} 13(3): 121-129. 

 http://www.science.uva.nl/$\sim$seop/entries/pascal-wager/
 

 Brams, Steven J. (1983) ``Superior Being, Their Powers, and the Problem of Evil:
Can This Be
Mathematics?'' {\it UMAP Journal,} 4(3): 265-283  (September 1983). 


Duncan, Craig (2003) ``Do Vague Probabilities Really Scotch Pascal’s Wager?'' {\it  Philosophical Studies}  112: 279-290 (February 2003)

 Durkin Jr., John T.; Greeley, Andrew M  (1991) 	``A Model of Religious Choice Under Uncertainty: On Responding Rationally to the Nonrational,''{\it  Rationality \& Society,}    3(2): 178-196 Apr91,.  

Isaac Levi (1982) ``A Note on Newcombmania,'' {\it Journal of Philosophy,} 79 (1982): 337-42. 


Richmond Campbell and Lanning Sowden (1985) {\it Paradoxes of Rationality and Cooperation: Prisoners' Dilemma and Newcomb's Problem,} (Vancouver: University of British Columbia Press, 1985).


	Richard M Gale 	{\it On the Nature and Existence of God,}
 Cambridge University Press
 	August 19, 1993



Montgomery, James D.   (1992) ``Pascal's Wager and the Limits of Rational Choice.
{\it 	Rationality \& Society,} ,   4(1): 117-121 (Jan92). 

Pascal, Blaise {\it Pensees}. 

Plato {\it The Republic}.

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