A man walks into a store. He selects a hat priced at $7 and gives the salesman a $10 bill. There is no change in the till, so
the salesman takes the bill over to the neighbor to break it. He comes back, gives the buyer the hat and $3 change. The next day the neighbor comes in and tells the salesman that the $10 bill he broke is counterfeit. The salesman takes a look at the bill and sees it is indeed a fake. He apologizes and gives the neighbor a new, genuine $10 bill. The question is: how much has the salesman lost in this triple transaction (assuming, for simplicity’s sake, that the price of the hat was equal to its cost)?
Over 50% of people (maybe you included), say the salesman lost $20. The math is easy: a $7 hat + $3 change + $10 given to the neighbor. Easy… But wrong. Try again: $7 hat + $10 given to the neighbor (remember not to add the $3 change because they belonged to the neighbor). The answer is therefore, 17$. The logic seems sound… But still, it is incorrect. Try and you might get: $13, $23, $10, $14 ($10 given to the neighbor + $7 for the hat - $3 change received from the neighbor).
If you happen to be located on the right side of the bell curve (approximately 15% of the population), you might very well have had the right answer all along - $10. If you are sure we are wrong, please have patience,, an explanation will soon follow. The thing is, there is a good chance you got the right answer using the “wrong” way.
We usually successfully deal with quantitative problems using process-oriented mathematical thinking. This type of thinking is deeply rooted in all our learning processes: we build a mathematical model that illustrates the question; solve the model using formulas and either brute force calculations or trial and error; or we use the a computer to quickly simulate the problem. The focus is on the process, thinking at the micro level (first X happens, then Y, etc.). The main advantage of this approach is that assuming you defined your analytical model properly and all the calculations are correct, you don’t have to think. As long as you do the donkeywork properly you will necessarily have the correct answer. The disadvantages are that such processes: require an accurate model and sound suppositions; they are lengthy and calculation is time consuming; chances of making a (silly) mistake in one of steps is directly related to the complexity of the process; and you can’t develop a “sense” about the correctness of the answer. Simply put, sometimes one “cannot see the wood for the trees.” An analytical thought process is not necessarily the logical path to the solution.
state-analysis thought processes
State-analysis thought processes offer an alternative approach. This approach is characterized by logical, rather than computational, thought and focusing on reaching conclusions that lead from the initial state (the problem) to the final state (the solution). You needn’t scrutinize the process, you can ignore the details and regard the case in toto at the macro level.
Back to the hat… let’s examine what happened to each of the characters (without going into detail). The neighbor didn’t gain or lose, as he was fully reimbursed. The customer gained precisely $10 (a $7 hat and $3 in change). Therefore… the salesman must have lost exactly $10. If you’re still not convinced, let’s try an alternative ending to our story: after giving the neighbor the $10 bill, the salesman took another look at the counterfeit bill and realized that they were wrong. The bill was, in fact, genuine. How much has he lost now??
Hummm… I’ll risk my reputation here.. lol
In the new scenario, the customer’s gain was 0 (a genuine bill for a $7 hat + $3 change); the neighbor’s gain was also 0 (he gave the salesman a genuine bill and the salesman compensated him for that same bill). Thus the salesman’s lost is 0 in this case, provided his bill examination process is 100% precise.
Interesting if we added some additional complexities, such as ethics, and others.
Thanks for the brain massage
hi Dov,
As a veteran (and admiring) student of yours, i was of course applying State Analysis as i was reading your post. my application of it i think was even more simple (maybe too simple? i think Roni H warned us from that in a post of his). there is one element here that is “doing the damage” and that is the bill. so any damage to anyone comes from the bill. and how much of a damage can a 10$ bill do? if its worth $0 then exactly $10, so thats the answer. using the same logic, if the bill is bona fide then no damage could have been done so the answer is 0. (true that you have to make sure that within the $10 or $0 damage there wasnt any specific damage to the salesguy but thata relatively easy).
thanks as always for a good learning
Amnon
Amnon hi
You are 100% right, and the more simple explanation you gave is exactly what I meant in the alternative end of the story. However, when applying state analysis one should be very careful building the chain of logic that takes him from the initial state (the problem) to the final state (the solution). For example when you state ” … and how much of a damage can a 10$ bill do?” and answer: “if its worth $0 then exactly $10, so that is the answer.” You are of course right here based on the fact that we are talking about Zero sum game, but assume a case of an armed robbery a 5 cent bullet may cause a considerable damage.
dov
Dov, if I understand correctly your answer to Amnon, then in a non-zero-sum game, the loss could be more or less than $10. For example, if we know that the salesperson makes a $2 profit on each hat he sells, then all he can really lose (maximum) from a counterfeit $10 bill in exchange for a hat is $8. The remaining $2 is simply a lack of potential profit that he could have gained. Right?
Yoni,
I dont agree with you, in our case he lost the 10$ not more and not less (in order to simplify the problem I defined that the cost of the hat and the price of the hat are the same) but even in a more complicted case that there is a profit, still you should calculate the lost of the potential profit and you will get this 10$.
in other words, from the state analysis point of view and based on the fact that in econimical transactions we are talking about zero sum game, if somebody gain 10$ than somebody has to lose the same 10$
And if we expanded our viewpoint so we thought that effects could be far in time and distance, and non proportional, the loss of reputation the counterfeit bill could generate to the multiple “givers” could be worth thousands times more than the intrinsic value of the bill. Adding on top of Dov’s bullet example, there might appear compunding effects such as the above mentioned negative word of mouth effect.
Cheers!
Fabian
If the hat cost $5, them the loss would be:
$10 - ($7-5) = $8
In Hong Kong, the hat might have cost $3. and been counterfeit also. So the loss would be minimizses