Truth or Envy? (a theory for prior-free mechanism design) Jason D. Hartline — Northwestern University (Joint work with Qiqi Yan) May 26, 2011
Truth
Mechanism Design Mechanism Design: how can a social planner / optimizer achieve objective when participant preferences are private. Challenge: designer does not know participant preferences, participants may strategize when reporting preference! 2 T RUTH OR E NVY ? – M AY 26, 2011
Incentive Compatibility Definition: a mechanism is incentive compatible (IC) if truthful reporting is an equilibrium. I.e., given others report truthfully, an agent maximizes her utility by reporting truthfully. 3 T RUTH OR E NVY ? – M AY 26, 2011
Incentive Compatibility Definition: a mechanism is incentive compatible (IC) if truthful reporting is an equilibrium. I.e., given others report truthfully, an agent maximizes her utility by reporting truthfully. Goal: Design IC mechanisms with good performance, e.g., profit. 3 T RUTH OR E NVY ? – M AY 26, 2011
Incentive Compatibility Definition: a mechanism is incentive compatible (IC) if truthful reporting is an equilibrium. I.e., given others report truthfully, an agent maximizes her utility by reporting truthfully. Goal: Design IC mechanisms with good performance, e.g., profit. Main Complication: IC constraints bind across preference profiles. 3 T RUTH OR E NVY ? – M AY 26, 2011
Incentive Compatibility Definition: a mechanism is incentive compatible (IC) if truthful reporting is an equilibrium. I.e., given others report truthfully, an agent maximizes her utility by reporting truthfully. Goal: Design IC mechanisms with good performance, e.g., profit. Main Complication: IC constraints bind across preference profiles. Consequences: • no mechanism is optimal for all preference profiles. • with prior distribution , can trade-off revenue. 3 T RUTH OR E NVY ? – M AY 26, 2011
Example Question: For • k units of an item, and • n agents with values drawn i.i.d. from U [0 , 1] what auction maximizes the seller’s expected revenue? 0 T RUTH OR E NVY ? – M AY 26, 2011
Example Question: For • k units of an item, and • n agents with values drawn i.i.d. from U [0 , 1] what auction maximizes the seller’s expected revenue? Answer: the k -item Vickrey auction with reserve 1 / 2 0 T RUTH OR E NVY ? – M AY 26, 2011
Example Question: For • k units of an item, and exponential with rate 1 • n agents with values drawn i.i.d. from U [0 , 1] what auction maximizes the seller’s expected revenue? Answer: the k -item Vickrey auction with reserve 1 / 2 0 T RUTH OR E NVY ? – M AY 26, 2011
Example Question: For • k units of an item, and exponential with rate 1 • n agents with values drawn i.i.d. from U [0 , 1] what auction maximizes the seller’s expected revenue? Answer: the k -item Vickrey auction with reserve 1 / 2 1 0 T RUTH OR E NVY ? – M AY 26, 2011
Example Question: For • k units of an item, and exponential with rate 1 • n agents with values drawn i.i.d. from U [0 , 1] what auction maximizes the seller’s expected revenue? Answer: the k -item Vickrey auction with reserve 1 / 2 1 Conclusion: optimal auction depends on prior distribution. 0 T RUTH OR E NVY ? – M AY 26, 2011
The trouble with priors The trouble with priors: 1 T RUTH OR E NVY ? – M AY 26, 2011
The trouble with priors The trouble with priors: • where does prior come from? 1 T RUTH OR E NVY ? – M AY 26, 2011
The trouble with priors The trouble with priors: • where does prior come from? • is prior accurate? 1 T RUTH OR E NVY ? – M AY 26, 2011
The trouble with priors The trouble with priors: • where does prior come from? • is prior accurate? • prior-dependent mechanisms are non-robust. 1 T RUTH OR E NVY ? – M AY 26, 2011
The trouble with priors The trouble with priors: • where does prior come from? • is prior accurate? • prior-dependent mechanisms are non-robust. • what if one mechanism must be used in many scenarios? 1 T RUTH OR E NVY ? – M AY 26, 2011
The trouble with priors The trouble with priors: • where does prior come from? • is prior accurate? • prior-dependent mechanisms are non-robust. • what if one mechanism must be used in many scenarios? Goal: theory for prior-free mechanism design. 1 T RUTH OR E NVY ? – M AY 26, 2011
The trouble with priors The trouble with priors: • where does prior come from? • is prior accurate? • prior-dependent mechanisms are non-robust. • what if one mechanism must be used in many scenarios? Goal: theory for prior-free mechanism design. (one of the main contributions of AGT to GT/Econ) 1 T RUTH OR E NVY ? – M AY 26, 2011
Envy
Multi-unit Pricing Problem: Multi-unit Pricing • n agents, values v 1 ≥ . . . ≥ v n • k units of an item. Goal: envy-free revenue-maximizing pricing. 3 T RUTH OR E NVY ? – M AY 26, 2011
Multi-unit Pricing Problem: Multi-unit Pricing • n agents, values v 1 ≥ . . . ≥ v n • k units of an item. Goal: envy-free revenue-maximizing pricing. First Attempt: • sell to top i at price v i gives revenue R ( i ) = iv i • pick j = argmax i ≤ k R ( i ) . 3 T RUTH OR E NVY ? – M AY 26, 2011
Multi-unit Pricing Problem: Multi-unit Pricing • n agents, values v 1 ≥ . . . ≥ v n • k units of an item. Goal: envy-free revenue-maximizing pricing. First Attempt: • sell to top i at price v i gives revenue R ( i ) = iv i • pick j = argmax i ≤ k R ( i ) . Note: can view as menu, a.k.a., pricing . 3 T RUTH OR E NVY ? – M AY 26, 2011
Example Example: suppose we have • 20 units of an item for sale, • 90 interested agents: – 10 with value 10, and – 80 with value 2. 4 T RUTH OR E NVY ? – M AY 26, 2011
Picture 180 100 40 10 20 90 5 T RUTH OR E NVY ? – M AY 26, 2011
Picture 180 100 40 10 20 90 5 T RUTH OR E NVY ? – M AY 26, 2011
Picture 180 100 40 10 20 90 Question: can we do better? 5 T RUTH OR E NVY ? – M AY 26, 2011
Example, revisited Example: suppose we have • 20 units of an item for sale, • 90 interested agents: – 10 with value 10, and – 80 with value 2. 6 T RUTH OR E NVY ? – M AY 26, 2011
Example, revisited Example: suppose we have • 20 units of an item for sale, • 90 interested agents: – 10 with value 10, and – 80 with value 2. Idea: price “randomized allocations”, a.k.a., lotteries. 6 T RUTH OR E NVY ? – M AY 26, 2011
Example, revisited Example: suppose we have • 20 units of an item for sale, • 90 interested agents: – 10 with value 10, and – 80 with value 2. Idea: price “randomized allocations”, a.k.a., lotteries. • Sell to all 10’s at price 9. • Sell to 2’s with probability 1/8 at price 2. • Revenue = 9 × 10 + 2 × 10 = 110 . 6 T RUTH OR E NVY ? – M AY 26, 2011
Example, revisited Example: suppose we have • 20 units of an item for sale, • 90 interested agents: – 10 with value 10, and – 80 with value 2. Idea: price “randomized allocations”, a.k.a., lotteries. • Sell to all 10’s at price 9. • Sell to 2’s with probability 1/8 at price 2. • Revenue = 9 × 10 + 2 × 10 = 110 . Is it envy-free? 6 T RUTH OR E NVY ? – M AY 26, 2011
Example, revisited Example: suppose we have • 20 units of an item for sale, • 90 interested agents: – 10 with value 10, and – 80 with value 2. Idea: price “randomized allocations”, a.k.a., lotteries. • Sell to all 10’s at price 9. • Sell to 2’s with probability 1/8 at price 2. • Revenue = 9 × 10 + 2 × 10 = 110 . Is it envy-free? • 10’s utility = 10 − 9 = 1 . • 10’s utility if swap = (10 − 2) × 1 / 8 = 1 . 6 T RUTH OR E NVY ? – M AY 26, 2011
Example, revisited Example: suppose we have • 20 units of an item for sale, • 90 interested agents: – 10 with value 10, and – 80 with value 2. Idea: price “randomized allocations”, a.k.a., lotteries. • Sell to all 10’s at price 9. • Sell to 2’s with probability 1/8 at price 2. • Revenue = 9 × 10 + 2 × 10 = 110 . Is it envy-free? Yes! • 10’s utility = 10 − 9 = 1 . • 10’s utility if swap = (10 − 2) × 1 / 8 = 1 . 6 T RUTH OR E NVY ? – M AY 26, 2011
Picture 180 110 100 10 20 90 7 T RUTH OR E NVY ? – M AY 26, 2011
envy-free pricing Definition: a pricing (and allocation) is envy free (EF) no agent wants to swap outcomes with another. 8 T RUTH OR E NVY ? – M AY 26, 2011
envy-free pricing Definition: a pricing (and allocation) is envy free (EF) no agent wants to swap outcomes with another. Main Simplification: EF constrains pricing outcome “pointwise”, nothing is required of pricing on different preference profiles. 8 T RUTH OR E NVY ? – M AY 26, 2011
envy-free pricing Definition: a pricing (and allocation) is envy free (EF) no agent wants to swap outcomes with another. Main Simplification: EF constrains pricing outcome “pointwise”, nothing is required of pricing on different preference profiles. Consequence: for any objective, there is an optimal envy-free pricing. 8 T RUTH OR E NVY ? – M AY 26, 2011
Truth or Envy(-free)? Thesis: envy freedom ≈ incentive compatibility 9 T RUTH OR E NVY ? – M AY 26, 2011
Truth or Envy(-free)? Thesis: envy freedom ≈ incentive compatibility Related Work: 1. IC = EF in limit [Jackson, Kremer ’07] 9 T RUTH OR E NVY ? – M AY 26, 2011
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