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A simple question for those familiar with argument analysis / formalization. Can you elaborate and explain what is going on in the following two arguments? They appear to lead to opposing conclusions but I'm not able to articulate the mechanics of each argument and how they compare and relate to one another.

ARGUMENT 1

you should buy something if and only if buying it implies a better situation than not buying it

buying lottery tickets implies having a chance of winning the lottery

not buying lottery tickets implies having no chance of winning the lottery

having a chance of winning the lottery is a better situation than having no chance of winning the lottery

conclusion: you should buy lottery tickets

ARGUMENT 2

you should buy something if and only if buying it implies a better situation than not buying it

buying lottery tickets implies losing 10 dollars

not buying lottery tickets implies keeping 10 dollars

keeping 10 dollars is better than losing 10 dollars

conclusion: it is not the case that you should buy lottery tickets

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  • Arguments with different premises lead to different conclusions. Commented Oct 12, 2023 at 23:36

3 Answers 3

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Both arguments have issues in that they focus on only the good or bad outcome in a situation where both are possible.

First, a basic assessment of key difference between each argument (no consideration of the soundness or completeness)

Argument 1

  • This argument focuses on the possible upside if you play and the denial of this upside if you don't play.
  • However, it ignores the possibility you will not win and therefore have less money (worse situation).

Argument 2

  • Premise 2 says you are guaranteed to not win, so it's obvious you shouldn't play the game.

That is the gist of why each comes to different conclusions.

Both arguments are valid, but not sound. They need additional premises to make the fundamental premise (the better/worse assessment) correct.

For argument 1, it's not clear why having a chance is better if we have to pay money for a ticket. In other words, there is an implicit assumption that makes the mere chance of winning better than the money they have to pay for it.

We can augment the arguments as follows:

**Argument 1.1 **

you should buy something if and only if buying it implies a better situation than not buying it

buying lottery tickets implies having a chance of winning the lottery

not buying lottery tickets implies having no chance of winning the lottery

The value I attach to the cost of the ticket is less than the value of the chance of winning the lottery.

Therefore, having a chance of winning the lottery is a better situation than having no chance of winning the lottery

conclusion: you should buy lottery tickets

Note: There are MANY things we buy that will not ever have a direct monetary payoff (e.g., food, movies) yet we deem them leaving us better off.

Argument 2.1

you should buy something if and only if buying it implies a better situation than not buying it

there is a very low probability of winning the lottery

(*) therefore, buying lottery tickets implies a high probability of losing 10 dollars and a very low probability of winning X dollars.

(*) not buying lottery tickets implies keeping 10 dollars and losing the chance to win X dollars.

keeping 10 dollars is more valuable to me than the small chance to win X dollars.

conclusion: it is not the case that you should buy lottery tickets

This conclusion is usually justified by appealing to the negative expected value of playing the lottery combined with a lack of entertainment value from gambling.

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It's easy. Both arguments fail, at line 4, to make a proper assessment of the outcome. Losing ten dollars is only better than not losing, all other things being equal. Having a chance of winning the lottery is only better than not having a chance, all other things being equal. You are just looking at individual choices in isolation and you are not considering the effect of a given choice on your other options.

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High volume gambling systems have very predictable "expected payouts"...

... meaning, it can be statistically calculated how much money will go out, versus how much money will come in.

In gambling situations such as blackjack and craps and roullette... the expected payout on a dollar bet is very high... just shy of 50%. The house sets the games up so they only win a little bit. Otherwise gamblers would quickly lose and leave.

Lottery tickets have much lower payout percentages. Typically about 20%. Meaning about 20% of the ticket sales will be returned as prizes..

Using this number... the calculated expected payout... we can predict what will be won based on what is bet... if spread out among all players.

And we can include it into the decision making.

For example...

If 10 million people pay $10 each in lottery tickets, the amount they each would win in return will be $2. AVERAGED OUT.

Some will come out ahead. Millions will not come out ahead. Most will not come out ahead (obviously).

One can, with calculated support, expect to lose money on the lottery ticket money spent. In almost all cases.


Noting. There can be exceptions. Some lotteries have cumulating jackpots. This can temporarily change their expected payout. Making them a more hopeful "investment".

But rarely will the situation be that the expected payout is over 100%. Lottery companies have good mathematicians.

There have been mistakes.

There is a movie about one lottery that structured their payout system poorly and it led to situations where "gambling enough was no longer gambling statistically speaking" and a man noticed it, and then involved his entire town in taking advantage of the situation.

https://en.wikipedia.org/wiki/Jerry_%26_Marge_Go_Large

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