Projects / How much to bid in a sealed auction when a win needs a later approval

How much to bid in a sealed auction when a win needs a later approval

Course project, Michigan Ross MBA. Decision tree with a win probability that depends on the bid, in Excel. Not client or employer work.

Excel

Excel modelDecision tree and the table of bids

Situation	A developer plans a sealed bid of $5M for a property. If the project goes ahead, it is worth $15M for $8M of capital.
Complication	A higher bid wins more often but leaves less profit. A win is also not the end: the plan then needs an approval that has a 30% chance, and without it the developer loses 10% of the bid.
Question	Which bid gives the highest expected value?
Answer	Bid $4.11M, not the planned $5M. The expected value rises 35%, but the bid wins only 12% of the time.

The bid moves two things in opposite directions: the chance to win and the profit of a win

Expected value of the bid$67k in the answer
- Chance to win
  - Bid amountThe lever. A higher bid wins more often
    Which bid is best?
    $4.11M. 12% chance to win; 20% at $5M.
- Expected result of a win
  - Profit if the plan is approved$15M less $8M of capital, less the bid
    How high can the bid go?
    $5.68M. Above this, the expected value is negative.
  - Chance of approvalFixed: 30%
    How low can it go before a $5M bid loses money?
    20%. The margin for error is 10 points.
  - Loss if the plan is not approvedThe limit: 10% of the bid

Lever what the decision changes Limit what the answer must respect Fixed outside the decision

Bid with the highest expected value

$4.11M

expected value $67k; $50k at the planned $5M bid

Chance to win at that bid

12%

20% at $5M: a higher bid wins more often but leaves less

Highest bid that is not a loss on average

$5.68M

above this, the expected value is negative

The expected value has its maximum at $4.11M and goes below zero above $5.68M

$3.0M$55kwins 6% of the time

$3.5M$63kwins 8% of the time

$4.0M$67kwins 11% of the time

$4.11M$67kwins 12% of the time

$4.5M$65kwins 15% of the time

$5.0M$50kwins 20% of the time

$5.5M$17kwins 26% of the time

$6.0M$-41kwins 34% of the time

Chance to win = 1 / (1 + e^(5 − 0.723 × bid)), bid in $ million. If the bid wins, the plan needs an approval (30% chance). With approval the project earns $15M less $8M of capital and the bid. Without it the bidder loses 10% of the bid.

Most of the time nothing happens; the gain comes from a 3% outcome

Outcomes of a $4.11M bid
Outcome	Probability	Result
The bid loses	88.4%	$0
The bid wins, the plan is not approved	8.1%	−$0.41M
The bid wins, the plan is approved	3.5%	+$2.89M

The model

Decision

The amount of one sealed bid for a property.

Objective

The highest expected value of the project.

Uncertainty

Two events in sequence: the bid wins or loses, then the plan is approved or not. The chance to win goes up with the bid.

With a fixed 20% chance to win, a $5M bid is worth $50k. When the chance to win depends on the bid, a lower bid of $4.11M is worth 35% more.
A $5M bid breaks even at an approval chance of 20%. The estimate is 30%, so the margin for error is 10 points.
The expected value is small against the stakes: $67k against a possible loss of $411k. A bidder who cannot take that loss must not bid.
The win curve is an assumption. The first thing to test is how the answer moves when the curve moves.