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Competition between Widget Manufacturers

By Jon Evans
Posted on Lifestyle

General Widget Makers (GWM) is in a fierce competition with United Widget Alliance (UWA). Every day each company can choose to either raise or lower the price of its widgets by 10 cents. By law:

  • The price cannot be unchanged from the previous day.
  • The price change must be fixed for the whole day.
  • The companies cannot collude or cooperate and must have no knowledge of what the price change for the other will be before both price changes are chosen for the day.

Neither company will ever exit the market, as each has determined the cost of withdrawal to far exceed the present value of any possible loss from participating into perpetuity.

Depending on the price changes for any given day, here is a table of the profit to GWM (equivalently the loss to UWA) for the day:

How would you recommend that GWM select its price changes over the next year to maximize its expected profit? Similarly, what would you recommend UWA do over the next year to maximize its expected profit?

Know the answer? Send your solution to ar@casact.org.

Truth versus Politics

One of nine candidates will lower taxes, and the others will either raise taxes or leave taxes unchanged. The candidate who will lower taxes tells the truth. Candidates who will raise taxes lie. The candidates, numbered 1 through 9, make the following statements:

  1. Either 3, 5, 7, 9 or I will lower taxes.
  2. I will leave taxes unchanged.
  3. Either 5 is telling the truth or 7 is lying.
  4. 1 is lying.
  5. Either 2 or 4 is telling the truth.
  6. 3 is lying.
  7. 1 is not going to lower taxes.
  8. I will raise taxes, and 9 will leave taxes unchanged.
  9. I will raise taxes, and 6 is lying.

You also know whether candidate 8 will leave taxes unchanged.

Bob Conger sent in the following solution. Begin with Candidates 8 and 9. For Statement 8 to be TRUE, 8a AND 8b must be TRUE (8a and 8b being the two component statements within 8). For Statement 8 to be FALSE, 8a AND/OR 8b must be FALSE. Similarly for statement 9.

Neither Candidate 8 nor Candidate 9 can be a tax-reducer, since tax-reducers tell the truth and each of these candidates claims to be a tax increaser.

If Candidate 8 were a tax no-changer, then Statement 8a would be False, meaning that Statement 8 would be False regardless of the True/False status of 8b. Thus, in this case the confidential information provided to us would not be sufficient to unravel the candidate mystery, in conflict with the characterization of the confidential information. Therefore, we conclude that:

  • Candidate 8 is a tax increaser, and therefore:
  • Statement 8 is False.

Statement 8 is False, but 8a is True. Therefore:

  • 8b is False, and
  • Candidate 9 is a tax increaser, and therefore
  • Statement 9 is False.

Statement 9 is False, but 9a is True. Therefore:

  • 9b is False, and
  • Statement 6 is True. Therefore:
  • Statement 3 is False. Therefore:
  • Candidate 3 is not a tax reducer.

Statement 3 is False. Therefore:

  • Statement 5 is False.
  • Statement 7 is True.

Statement 7 is True. Therefore:

  • Candidate 1 is not a tax reducer.

Statement 5 is False. Therefore:

  • Candidate 5 is not a tax reducer.
  • Statement 2 is False.
  • Statement 4 is False.

Statement 2 is False. Therefore:

  • Candidate 2 is a tax increaser.

Statement 4 is False. Therefore:

  • Statement 1 is True. Therefore:
  • Either 1, 3, 5, 7, or 9 is a tax-reducer.

Therefore:

  • Candidate 7 is the tax reducer.

Solutions were also submitted by Kyle Bartee, Brock Childs, Sam Kessler, Billy Litner, Juan C. McNamara and Brad Rosin.

AR Puzzle Editor Jon Evans is president of Convergent Actuarial Services, Inc. in Delray Beach, Florida.