A designer builds an auction with a reserve price so high that bidders with low valuations earn negative expected payoff from participating. What happens, and which constraint is violated?
ALow-valuation bidders participate anyway because the reserve price signals quality; no constraint is violated
BLow-valuation bidders opt out; the individual rationality (participation) constraint is violated for their type
CLow-valuation bidders misreport their type to avoid the reserve; the incentive compatibility constraint is violated
DThe mechanism still works because the designer only needs high-valuation bidders to participate
The IR constraint requires that each type earns at least their reservation utility (usually zero) from participating. If a low-valuation type expects negative payoff, they will simply not show up — their outside option (not participating) is better. This is an IR violation, not an IC violation. IC governs whether participants *report truthfully*; IR governs whether they *show up at all*. Option D is tempting but wrong: a mechanism where entire type-ranges drop out is unraveling, not functioning.
Question 2 Multiple Choice
Which version of the individual rationality constraint is *hardest* for a mechanism to satisfy, and why?
AEx ante IR, because agents must accept the mechanism before knowing their type
BInterim IR, because it must hold for every possible realization of other agents' types
CEx post IR, because every agent must be happy with the outcome after observing all information
DAll three are equally binding since they all require nonnegative expected utility
Ex post IR is the strongest requirement: every agent must be at least as well off as their outside option *after* the full outcome is realized, regardless of how unlucky they were. This rules out mechanisms where some types sometimes 'lose' from participation even if they expected to gain on average. Interim IR (the standard in Bayesian design) only requires nonnegative expected payoff given the agent's own type; ex ante IR only requires nonnegative payoff before knowing even one's own type. Stronger IR constraints shrink the feasible mechanism space — ex post IR rules out many efficient mechanisms that interim IR permits.
Question 3 True / False
In the optimal auction, the IR constraint for the lowest-type agent directly pins down the informational rents that must be paid to higher-type agents.
TTrue
FFalse
Answer: True
This is the core interaction between IR and IC in mechanism design. To prevent high-valuation bidders from mimicking low-valuation ones (IC), the mechanism must give high types a payoff premium — an informational rent. The IR constraint for the lowest type (which must hold with equality in the optimal mechanism to extract maximum revenue) serves as the 'floor.' IC constraints then require that higher types receive weakly more surplus than lower types, creating a ladder of rents. The designer cannot both satisfy IR at the bottom and IC throughout while extracting all surplus from high types.
Question 4 True / False
The individual rationality constraint is mainly relevant in environments where agents have private information about their types.
TTrue
FFalse
Answer: False
IR is required whenever participation is voluntary, regardless of whether there is private information. Even under complete information, a mechanism must leave each participant at least as well off as their outside option — otherwise they simply refuse to participate. The IR constraint captures the fundamental fact that any voluntary institution must compete with the option of not participating. Private information makes IR harder to satisfy (since the designer doesn't know each agent's true reservation value) but is not what gives rise to the constraint in the first place.
Question 5 Short Answer
Why does the IR constraint force a mechanism designer to leave 'informational rents' to high-type agents, even when the designer would prefer to extract all surplus?
Think about your answer, then reveal below.
Model answer: The designer must satisfy IR for the lowest type (they receive zero expected surplus) and IC for all types (each type prefers reporting truthfully to mimicking a lower type). IC requires that high types earn strictly more than they would receive if they reported as a lower type. Since the lowest type gets zero (IR), each step up in type must come with additional surplus to deter downward misreporting. This creates a chain: high types accumulate informational rents simply because they could credibly claim to be a lower type and still be better off — the designer must compensate them for being honest.
The intuition is that private information gives high-type agents 'leverage': they know something the designer doesn't and can exploit it. The only way to get truthful reporting (IC) from high types is to reward them for revealing their type. The IR constraint for the lowest type sets the baseline at zero; every other type must receive weakly more, and this strictly positive payoff for high types is the 'rent' that private information generates. In auctions, this is why the seller never captures all the surplus even in the optimal mechanism.