Gambling in the US is more popular than at any time this century. State lotteries are mushrooming all over. Las Vegas has been cloned to produce Atlantic City. Riverboat gambling has returned to the Mississippi. And the usual betting on horses, cards, sports and numbers has increased.
Though one lottery ad tells us "You have to play to win,'' you certainly don't have to play to lose! Not only do millions of people spend billions of dollars every year on losing lottery tickets, but those who live in states with legalized gambling also pay for its costly effects on the social fabric, whether they gamble or not.^{1}
But what most of us don't realize is that all of us are gambling with far higher stakes than these  we are gambling with our lives! The French mathematician Blaise Pascal (16231662) was perhaps the first to recognize this. In his famous Penseés, Pascal noted out that everyone either lives like God exists or like He doesn't.^{2} Since the God of the Bible is invisible, infinite and incomprehensible, none of us humans can be quite certain He doesn't exist. There is, in fact, more than adequate evidence that He does exist.^{3} But if we demand absolute proof before we will give up our selfcentered lifestyles, this level of proof will not arrive before we die or Jesus returns  in either case, too late to hedge our bets.
Thus, there is an important sense in which each of our lives is a gamble, wagered on the truth or falsity of the proposition "God exists." Like most wagers, the outcome is subjectively uncertain. We stake our lives, which we certainly have, against an uncertain gain or loss, which depends both upon our own choice and the outcome of the bet. Unlike the usual bet, however, this one is not optional.
The Theory of Games
In this century Pascal's work of applying probability theory to games of chance has been extended to more complicated situations in real life, including investments, diplomacy and warfare, where the probability of a particular outcome is unknown in advance. A good introductory article on this subject appeared in the December, 1962, issue of Scientific American.
To help us understand Pascal's wager, let us consider one of the simplest
problems in the mathematical theory of games, the twobytwo matrix game.
A matrix is merely a collection of numbers in a particular order. A twobytwo
matrix is a collection of four numbers (represented here by the letters
a, b, c, and d) arranged so that a and b form one row, and c and d another
row below it:




Thus a and b are called "row one," c and d are called "row two." Similarly a and c are "column one," b and d are "column two."
A matrix game involves two players, say Ron and Charles. Ron will secretly choose one of the two rows, while Charles covertly selects one of the two columns. The selections are written on slips of paper and handed over to the judge, who opens both and announces the result for that play of the game, a particular row and column, and the number corresponding to that location. For instance, if Ron chose row one and Charles column two, the result would be b. If b is positive, Ron wins the value of b (in dollars, say) from Charles; if b is negative, Ron loses the same amount to Charles. Sometimes the game will consist of a number of plays, the appropriate payoff being made each time. In other cases, such as warfare, there may only be one play to end the game.
The character of the particular game being played depends crucially
on the relative value of the numbers in the matrix. Some games are rather
boring. If, say, the matrix is:




then Ron will always win if he plays row one. Charles will soon quit if he can even be convinced to play in the first place. Of course, if the game is a war, Charles may have no choice.
A more complicated game has both a postiive and a negative number in
each row and column, for example:




Now if Ron consistently plays the first row, Charles will eventually settle on the second column and Ron will lose his shirt. But if Charles begins to play the second column consistently, Ron can switch to the second row and begin winning again. Under such circumstances, both players will usually adopt some strategy that plays first one row or column and then another. Perhaps Ron will play row one a fraction p of the time, and row two the rest of the time (a fraction 1  p). Charles' strategy, in the absence of a spy network to find out what Ron is doing, will be to play column one a fraction q of the time, otherwise column two.
Ron's expected winnings may be calculated simply by taking a weighted average of the four possible outcomes, multiplying each outcome by the fraction of the time it will occur as Ron and Charles pursue their particular strategies. The equation for Ron's expected winnings E is:
E = pqa + p(1q)b + (1p)qc + (1p)(1q)d.
For the second matrix game above, we plug in a = 1, b = 4, c = 3 and d = 1. With a little algebra, we get:
E = 9pq  5p  4q + 1.
Now if Ron never plays row one (p = 0) and Charles never plays column one (q = 0), then E = 1, and Ron will win one dollar every time the game is played. If however, p = 0, Charles may always play column one (q = 1), and then E = 3 and Ron loses $3 every play. Ron's best strategy is to choose p so as to make E as large as possible no matter what Charles does. Charles' best strategy is to choose q so as to make E as small as possible, independent of what Ron does. Though beyond the scope of our simple treatment here, you may be interested to know that Ron's best strategy is to play row one 4/9ths of the time. Charles' best strategy is to play column one 5/9ths of the time. The value this gives for E is 11/9ths, so that Ron will lose about $1.22 on each play. Ron should not play this game at all if he can avoid it!
Application to Pascal's Wager
We can set up Pascal's wager in the form of a twobytwo matrix game.
Let the first column represent the case that Christianity is true and the
second that Christianity is false and there is no survival after death
(the usual secular view). Let row one represent acceptance of Christianity
and row two rejection of Christianity.
Christianity:  True  False 
Accepted  a  b 
Rejected  c  d 
Then player Ron is any living person  you, I, or someone else  who must either live as though Christianity is true or as though it were false. Player Charles is Reality, the Grim Reaper, Chance, God, Eschatological Verification, or something of the sort, which will eventually reveal to each individual the wisdom or folly of his or her choice.
What are the values for the numbers a, b, c, and d? The relative values of these numbers will be crucial to determine what type of game we are playing and what our strategy should be.
Consider first the value of d, the payoff if Christianity is false and has been rightly rejected. Taking the alternative here to be some sort of secular humanism with no survival after death, the payoff has been collected before death. It will vary widely in value, depending on the type of life the person experienced. Let us choose the maximum value which might be obtained  perhaps a long pleasant life of health, wealth and happiness. In any case, this value will be a finite number, so we will set it equal to one, thus making the value of such a life the unit in our system of rewards. Thus d = 1.
Now consider the values of a and c, the payoffs in case Christianity turns out to be true. I am using "Christianity" in the sense defined in the Bible, so these values may be derived from the 25th chapter of Matthew's Gospel, where Jesus at the judgment says to believers, "Come, you who are blessed by my Father; take your inheritance, the kingdom prepared for you since the creation of the world." These go with Him "into life eternal" (verses 34, 46). To the unbelievers He says, "Depart from me, you who are cursed, into the eternal fire prepared for the devil and his angels." These go "into everlasting punishment" (vv 41, 46). Thus the value of a is positive and infinite, whereas that of c is negative and infinite.
In regard to the value of b, it appears that Pascal was mistaken in
thinking that nothing has been lost if one should live as if Christianity
were true but be mistaken, i. e., that b = 0. The apostle Paul, with a
sober recognition of the facts of persecution, says, "If only for this
life we have hope in Christ, we are to be pitied more than all men" (1
Cor 15:19). But even if Christianity is false and there is no survival
after death, this is only a finite loss, though the Christian should fare
worse than anyone else playing the game. For simplicity, let b = 1.
Christianity: 


Accepted 


Rejected 


Now for the strategy. Player "Charles" is reality. His play is either column one, "Christianity is true," or column two, "Christianity is false." Naturally Christians will think Reality always plays column one; atheists that it always plays column two. But if we are consistent empiricists (or rationalists who admit we cannot rule out the possibility that Christianity may be true), then we must allow at least some small probability e that column one will be played. Thus q = e.
Ron must now decide on his optimum strategy. Since he personally has only one play, he must choose either to live like Christianity is true or to live like it is false. However, as there may be others who will take his advice, will Ron advise: (1) all men to accept Christianity, (2) all to reject Christianity, or (3) some to accept and some to reject? Will Ron choose his strategy to be (1) p = 1, (2) p = 0, or (3) p = some intermediate value?
Ron's expected winnings are given by the formula below, as N is allowed to go to infinity:
E = peN  (1p)eN  p(1e) + (1p)(1e)
Now since e is positive and not equal to zero (no matter how small it is), as N gets larger and larger, the first term peN becomes large and positive, the second term (1p)eN large and negative, and the remaining terms trivially small by comparison. Ron can thus make his expected winnings as large as possible by choosing p = 1, so that the second term becomes zero.
Ron, therefore (and you and I), should himself live as though Christianity is true and also advise others to do the same.
Generalization of Pascal's Wager
But, we may object, isn't this too simple? There are certainly more than two religions or philosophies in the world. What about Hinduism, Islam, and the various New Age religions? Don't they count? Let's see.
Pascal's wager may be generalized by expanding it into a choice among n different worldviews. The resulting version in modern game theory involves an nbyn matrix. The diagrams and arithmetic are rather more complicated than before, but I have set this out in some detail elsewhere.^{4} The upshot is that Ron's optimum strategy is to select only some combination of those worldviews which have both an infinite heaven and hell, and provide no additional lives in which to guess again. I am no expert in comparative religions, but I suspect Christianity has few competitors in this regard.^{5} Most of the heresies which have developed from Christianity have sought to remove the doctrine of eternal punishment also.
What's Your Bet?
Pascal's wager continues to have its original force in its twochoice version, even centuries after he first formulated it. This force is somewhat diluted in the multiplechoice version, but only relative to religions featuring both heaven and hell. Among such religions Pascal's wager has nothing to say, and a choice must be made on other grounds, hopefully on the basis of evidence.^{6}
Many unbelievers (and some believers) have been put off by the alleged low morality of Pascal's wager. But the argument is not intended to be a moral argument, rather a prudential one. It merely reminds us that it is stupid to go though life without investigating religions in which the stakes are infinite.
Remember the advice of Jesus: "As you are going with your adversary to the magistrate, try hard to be reconciled to him on the way, or he may drag you off to the judge, and the judge turn you over to the officer, and the officer throw you into prison. I tell you, you will not get out until you have paid the last penny'' (Luke 12:5859).
Written by Robert C. Newman, PhD, astrophysics, Cornell
References
1. Tom Watson, Jr., Don't Bet on It (Baker, 1987).
2. Blaise Pascal, Penseés, section III.
3. See, for example, John Warwick Montgomery, ed., Evidence for Faith: Deciding the God Question (Probe/Word, 1991); Robert C. Newman, ed., Evidence of Prophecy (IBRI, 1988); Charles B. Thaxton, Walter L. Bradley and Roger L. Olsen, The Mystery of Life's Origin (Philosophical Library, 1984).
4. Robert C. Newman, "Pascal's Wager Reexamined," Bulletin of the Evangelical Philosophical Society 4 (1981): 6167.
5. Orthodox Islam has a heaven and hell, but admits Christians as "people of the Book.'' Biblical Christianity cannot return the favor, as Jesus said, "No one comes to the Father except through me'' (John 14:6).
6. Besides those in note 3, see Kenny Barfield, Why the Bible is Number 1: The World's Sacred Writings in the Light of Science (Baker, 1988).