La mujer y la violencia de género en los medios de comunicación impresos impresos

In this section I compare the ex-post welfare of the standard equilibrium of the fixed contribution mechanism, as described in section 2.6, and a standard equilibrium of the additional contribution mechanism as described in section 2.7. The central result (Proposition II.22) is that the additional contribution mechanism performs at least as well as the fixed contribution mechanism on any type space, and performs better on some type spaces. In the terminology introduced in section 2.4, this means that any standard equilibrium of the additional contribution mechanism will improve on the standard equilibrium of the fixed contribution mechanism.

Our comparison will be between the fixed contribution mechanism with initial shares s⁰ and the additional contribution mechanism with the same initial shares s⁰, with the corresponding standard equilibria. The criterion will be the improvability relationship, as defined in section 2.4.

Proposition II.18. For any type space T and type profile t, and given the standard equilibria, the additional contribution mechanism implements the social welfare max-imizing outcome if the fixed contribution mechanism implements the social welfare maximizing outcome.

Proof. There are two cases to consider: when welfare maximization requires that the public good not be created at t, and when welfare maximization requires the public good to be created.

When welfare maximization requires the public good not be created, the sum of valuations is less than c, so there is no possible ˆs such that for all i, ˆs_i ≤ ˆv_i(t_i).

Therefore in a standard equilibrium of the additional contribution mechanism, at least one agent must choose “No” and the public good will not be created. Likewise it must be that s⁰_i > v_i for some i, guaranteeing that the public good is not created under

the standard equilibrium of the fixed contribution mechanism. So neither mechanism will produce the good.

When welfare maximization requires the public good be created, we need to show that if the fixed contribution mechanism leads to the public good being created, then so does the additional contribution mechanism. Recall that I defined a standard equilibrium of the additional contribution mechanism as having the following property (equation (2.21)):

ˆ

v(t) ≥ s⁰ ⇒ q^M(a^∗_T(t)) = 1

for all t ∈ T and all T ∈ Ω, which ensures that whenever the standard equilibrium of the fixed contribution mechanism produces the public good, so does the standard equilibrium of the additional contribution mechanism.

Therefore both when producing the public good is efficient and when it is not effi-cient, the standard equilibrium of the additional contribution mechanism maximizes welfare if the standard equilibrium of the fixed contribution mechanism maximizes welfare.

Proposition II.19. There exist T containing type profiles t where, for the standard equilibria of the additional contribution mechanism, the additional contribution mech-anism implements welfare maximization but the fixed contribution mechmech-anism, with its standard equilibrium, does not.

Proof. To demonstrate the existence of type spaces that satisfy this condition, I use the following result that only depends on the primitives of T. The logic here is that if agent 1 has a valuation above her initial share, knows the true valuations of the other agents with probability one, and those agents all have valuations at or below their initial shares (with at least one strictly below), then agent 1 can ensure the

public good is created by choosing appropriate additional contributions to direct to the other agents.

Lemma II.20. For any type profile t such that

•

any standard equilibrium of the additional contribution mechanism implements the welfare maximizing outcome (creating the public good) at t.

Proof. Given agent 1’s beliefs, agent 1 will only expect the public good to be created if and only if for all i 6= 1, bi ≥ s⁰_i − ˆvi(ti). The minimum amount of additional contribution that will lead to the public good being created is then

N

X

i6=1

s⁰_i − ˆv_i(t_i)

If agent 1 makes that additional contribution, she receives a payoff of

ˆ

Agent 1 gets a payoff of 0 if the public good is not created, so she is willing to pay the minimal additional contributions that ensure the public good gets created.

Therefore, for all i > 1

b_i = s⁰_i − ˆv_i(t_i)

and for all i, ˆs_i ≤ ˆv_i(t_i) which implies the public good is created.

Let T be any type space. Then the above lemma leads immediately to the follow-ing sufficient condition:

Lemma II.21. If there exists a t⁰ ∈ T that satisfies the conditions of Lemma II.20, then at t⁰ ∈ T the additional contribution mechanism and its standard equilibria implement the welfare maximizing outcome (the public good is created), while the fixed contribution mechanism and its standard equilibrium do not.

Proof. Lemma II.20 implies that the additional contribution mechanism implements the welfare maximizing outcome. The fixed contribution mechanism, however, will not lead to the public good being created, because there is some j such that ˆv_j(t_j) < s⁰_j, so the fixed contribution mechanism does not implement the welfare maximizing outcome.

Notably, the universal type space satisfies the conditions of lemma II.20, as do some finite type spaces. So if Ω includes the set of all finite type spaces or the universal type space then lemma II.21 will hold for some T ∈ Ω.

Propositions II.18 and II.19 together give us our result:

Proposition II.22. If Ω includes at least one T that satisifies the conditions of lemma II.20, then: for any shares s⁰, the additional contribution mechanism with

initial shares s⁰ and its standard equilibrium improves on the fixed contribution mech-anism with fixed shares s = s⁰ and its standard equilibrium.

Proof. Proposition II.18 establishes that for every T and every t ∈ T , the additional contribution mechanism is at least as efficient. Proposition II.19 establishes that for some T and t ∈ T , the additional contribution mechanism is strictly more efficient.

These two claims together establish that the additional contribution mechanism im-proves on the fixed contribution mechanism.

Combined with Proposition II.16, which shows that any feasible dominant strat-egy mechanism is either improved on or equivalent (from an efficiency standpoint) to a fixed contribution mechanism, Proposition II.22 implies that every feasible domi-nant strategy mechanism is improved on by some additional contribution mechanism on any Ω with a T that satisfies the conditions of lemma II.20. In particular, if Ω includes the set of all finite type spaces or the universal type space, then every fea-sible dominant strategy mechanism is improved on by some additional contribution mechanism.