Who wants a dollar

[BMAD]

Suppose you and a logical but competitive friend are at a $1 auction.  The auction works as such, you each start the contest with $.50 and you bid on $1 bills.  The highest bid for each round wins $1 minus the amount bid AND loses their actual bid.  So for example, if in round 1, you bid $.20 and win the $1 then your new balance would be $.50 - 0.20 + (1-0.20) = 1.10.  If a tie occurs, the amount gained for each individual (i.e. 1 - bid) is divided by 2 but the full bid is still subtracted from the totals. Lastly $0 bets are not allowed.

Knowing that your friend is competitive and logical...

how much should you bid?

what would be your expected payout after five rounds?

 

[BMAD]

Let me add that if you are out bid,  you lose the  money that you chose to bet. 

Edited June 17, 2016 by BMAD

[Rollie]

Spoiler

I will bet $0.50 the first round. If my friend bets anything but $0.50, he will lose and I will win the rest by simply bettering $0.51, so he is guaranteed to also bet $0.50. Each round, we will both bet this, pay it, and receive $0.50 back, keeping our totals the same.

 

[BMAD]

1 hour ago, Rollie said:

  Hide contents

I will bet $0.50 the first round. If my friend bets anything but $0.50, he will lose and I will win the rest by simply bettering $0.51, so he is guaranteed to also bet $0.50. Each round, we will both bet this, pay it, and receive $0.50 back, keeping our totals the same.

 

Logically you would both bid the same amount then meaning.... 

.50 -. 50 + (1-.5)/2 =. 25

So your strategy would cut your winnings by 50%.

Is this the best option? 

[Rollie]

Ohh, misunderstood  In that case, I don't think there is a real answer for this. A game theory approach would say we both continuously bet $0.01 every round, ensuring we rack up $0.24 each per round. However, it's been stated my friend is competitive, so knowing this, he will want to win, not just draw for mutual benefit. So he will bet $0.02, to get all the money, and then just outbid me for every subsequent round. Knowing this, I could say I will then bet $0.03 instead, to which he would deduce he should bet $0.04. This continues until we both realize we should bet $0.50, a price at which we both realize we will lose money, but realize we will lose the competition if we ever bet less than our total savings; so .25 after one round, and then it would stay at .25 because we would continue to all-in for a payout of 0.25.

If my friend was even more cooperative and trustworthy, we would alternate games winning, with one person winning each round, maximizing mutual gain.

 

[BMAD]

On 6/17/2016 at 2:55 PM, Rollie said:

Ohh, misunderstood  In that case, I don't think there is a real answer for this. A game theory approach would say we both continuously bet $0.01 every round, ensuring we rack up $0.24 each per round. However, it's been stated my friend is competitive, so knowing this, he will want to win, not just draw for mutual benefit. So he will bet $0.02, to get all the money, and then just outbid me for every subsequent round. Knowing this, I could say I will then bet $0.03 instead, to which he would deduce he should bet $0.04. This continues until we both realize we should bet $0.50, a price at which we both realize we will lose money, but realize we will lose the competition if we ever bet less than our total savings; so .25 after one round, and then it would stay at .25 because we would continue to all-in for a payout of 0.25.

If my friend was even more cooperative and trustworthy, we would alternate games winning, with one person winning each round, maximizing mutual gain.

 

I don't think this works though. Your friend being logical and competitive could just bet 1 cent and lose the first round and by you winning on 25 cents he/she would have more than you at 49cents and then could forever dominate the auctions. 

[Rollie]

But given that the prize is only cut in half if there is a tie, if during round one, I bet 0.50, and he bets 0.01, I get a prize of 0.50, leaving me with 0.50. He however forfeits his 0.01 bet, leaving him with 0.49.  As such, I'm guaranteed to win the next bet as well, which will bring him down to 0.48, and leaving me with 0.50. At this point, I can start lowering my bid 0.01 each round, increasing my earnings each time.

So the results of the 0.50/0.01 betting scheme

My Money    Friend's Money  My Bet   Friend's bet

0.5                0.5                      0.5        0.01

0.5                0.49                    0.5        0.01

0.5                0.48                    0.49      0.01

0.52              0.47                    0.48      0.01

0.56              0.46                    0.47      0.01

0.62              0.45                    0.46      0.01

(my win)

 

Results of 0.5/0.5 betting scheme (recalculated after looking yet again at the problem description more carefully)

My Money    Friend's Money  My Bet   Friend's bet

0.5                <same>              0.5        <same>

0.25              <same>              0.25      <same>

0.37              <same>              0.37      <same>

0.31              <same>              0.31      <same>

0.34              <same>              0.34      <same>

0.33              <same>              0.33      <same>

0.33              <same>              0.33      <same>  (repeats forever)

(tie)

[CaptainEd]

Rollie, your friend is "logical" and "competitive". But is your friend "greedy" as well?
Does your friend prefer making more money than you? or more money, period?
You suggested cooperating with your friend
what if you...
 

Spoiler

both bet a penny each time

my holdings	friend holdings	my bet	friend bet
$0.50	$0.50	$0.01	$0.01
$0.99	$0.99	$0.01	$0.01
$1.47	$1.47	$0.01	$0.01
$1.96	$1.96	$0.01	$0.01
$2.44	$2.44	$0.01	$0.01
$2.93	$2.93

I think BMAD wants to introduce us to the fact that there can be an outcome that is better for both players than the "logical", "competitive", "rational" outcome.
Thanks, BMAD!

 

[BMAD]

1 hour ago, CaptainEd said:

Rollie, your friend is "logical" and "competitive". But is your friend "greedy" as well?
Does your friend prefer making more money than you? or more money, period?
You suggested cooperating with your friend
what if you...
 

  Hide contents

both bet a penny each time

my holdings	friend holdings	my bet	friend bet
$0.50	$0.50	$0.01	$0.01
$0.99	$0.99	$0.01	$0.01
$1.47	$1.47	$0.01	$0.01
$1.96	$1.96	$0.01	$0.01
$2.44	$2.44	$0.01	$0.01
$2.93	$2.93

I think BMAD wants to introduce us to the fact that there can be an outcome that is better for both players than the "logical", "competitive", "rational" outcome.
Thanks, BMAD!

 

Exactly

[Rollie]

3 hours ago, CaptainEd said:

Rollie, your friend is "logical" and "competitive". But is your friend "greedy" as well?
Does your friend prefer making more money than you? or more money, period?
You suggested cooperating with your friend
what if you...
 

  Hide contents

both bet a penny each time

my holdings	friend holdings	my bet	friend bet
$0.50	$0.50	$0.01	$0.01
$0.99	$0.99	$0.01	$0.01
$1.47	$1.47	$0.01	$0.01
$1.96	$1.96	$0.01	$0.01
$2.44	$2.44	$0.01	$0.01
$2.93	$2.93

I think BMAD wants to introduce us to the fact that there can be an outcome that is better for both players than the "logical", "competitive", "rational" outcome.
Thanks, BMAD!

 

That goes back to my previous path of reasoning, which briefly described the $0.01 scenario (i.e., if my friend believes I will bet 0.01, he can bet 0.02 and win everything from then on without my cooperation). It depends if my friend is trustworthy, and also whether competitive means making as much more than me as possible, maximizing their own profit, or simply guaranteeing they have funds >= to mine.  It also depends whether my friends thinks that I am trustworthy, because even if he's willing to do the 0.01/0.01 trade over and over, if he thinks I'm going to betray him, he will follow the same reasoning as I had above, leading to the previously mentioned 0.50/0/50 battle.

[BMAD]

hmmm.... you have a point rollie.