−4 −4acusar
Lliçó 3. Funció de demanda d’un bé d’un consumidor preu acceptant
Proof of Proposition 1
Proof. First, note that +#1= %% = +#, = +0 is not an equilibrium as soon as ( 4 ) " 1 since each player has incentives to deviate to strategy +1 with a strictly positive payo!
of ) " 1 " (.
Assume s# = (+#1$ %%+#,) is an equilibrium, and player ' wins. This implies that all his rivals play +#% = +0, +@ ! # : @ 6= ' (otherwise rivals of ' obtain negative payo!). If all rivals of ' play +0, player ' maximizes his payo! by playing +#! = +1, which is his best response. If player ' plays +1 and other players play +0 then some player @ 6= ' can win with * = 2 by playing strategy +% = +3, which yields a payo! of 7% = ) " 2 " 2(.
If ( 4 *2 " 1 then player @ proÞtably deviates from +0 to +3 in which case the best response of ' is to play +#! = +0. Now all rivals of @ play +0, and hence the best response of @ is +#% = +1 but as just has been shown for player ', strategy combination
¡+#%$ s#!%¢
= (+1$ +0%%+0) is not an equilibrium. Therefore no Nash equilibrium in pure strategies exists if ( 4 *2 " 1. If ( $ *2 " 1, no player @ has incentives to deviate from +0, and hence strategy combination (+#!$ s!!) = (+1$ +0%%+0) is an equilibrium.
Proof of Lemma 2
Proof. In "2 the strategy set is ,2 = {+0$ +1$ +2$ +3} = {(00) $ (10) $ (01) $ (11)}. Let
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12 = (102$ 112$ 122$ 132) be a symmetric mixed strategy equilibrium in "2. For an arbitrary player ' expected payo!s are
3!¡
The latter is due to Lemma A.2. For easy reference, we repeat the list of all relevant win probabilities: or both, and determine the equilibrium mixed strategy.
CLAIM 1. 10200.
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We will consider the following four exhaustive cases:
(i) 3!(+2$ 12) = 0$ 3!(+3$ 12) = 0 and 122$ 0$ 132$ 0%
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In the latter derivation recall that 8 (&) 4 1 for any & $ 3.
(iii) From 132 = 0, we obtain 112 = 1 " (102+ 122). Substituting from Equation (B-1) 102 + 122 = # !1p
-*!1 yields 112 = 1 " # !1p
-*!1 0 0. Condition 3!(+2$ 12) = () " 2) 22(+2$ 12) " ( = 0 yields (102+ 112)$ !1" (& " 1) 112(102)$ !2 = *!2- . Substitut-ing for 112 results in the equation
µ which implicitly deÞnes b10()$ ($ &). The left-hand side positively depends on b10()$ ($ &).
To see this, substitute the binomial expansion of (102+ 112)$ !1 =$ !1P
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with the latter due to
)#" 2
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hence e1+ is a well deÞned probability distribution with support on ,+. We will show that e1+ is equilibrium in "+ = ¡#$ ,+${7!}!",¢
. Assume player ' plays some 10++1 in
"++1 and his expected payo! is therefore 3!
¡10++1$ 1++1
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Equilibrium payo! of player ' is3!(1++1$ 1++1) = 3#(1++1) ·X It only remains to note that any 10+ ! ! (,+) is a special case of probability distribution from ! (,++1), as it assigns probability of zero to all strategies + such that + (6 + 1) = 1.
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It follows that 3!(e1+$e1+) $ 3!(10+$e1+), for any 10+ ! ! (,+) and thus e1+ is equilibrium in "+.
Proof of Proposition 3
Proof. Consider three types of equilibrium in "2in Lemma 2. From the proof of the lemma, equilibrium (iii) requires 3!(+3$ 12) % 3!(+2$ 12), or, equivalently, 22(+3$ 12) % 22(+2$ 12), which holds if and only if 102$ (& " 1) 112, i.e 102+ 112$ &112. At the same time, 132 = 0 implies 122 = 1 " (102+ 112). Combining the two yields 122 % 1 " &112 = 1 " &³
1 " # !1q
1
*!1
´
. From 122 $ 0 we obtain the necessary condition for equilibrium (iii):
1 " &
µ
1 " # !1
r (
)" 1
¶
$ 0 ) # !1
r (
)" 1 $ 1 " 1
& ) (·
µ &
& " 1
¶$ !1
$ ) " 1 ) ) % 1 + ( ·1 + 8 (&) 8(&) %
Assumption ( 4 () " 1)1+.($ ).($) violates this necessary condition and rules out equilibrium (iii).
A necessary condition for equilibrium (ii) is 91 $ 0, i.e. ) % 2!.($)1!.($), which, combined with the above, implies 2!.($)1!.($) 0 1 + ( · 1+.($ ).($) , equivalent to 1!.($)1 0 (· 1+.($).($) , or 8 (& ) 0 ( · (1 " 82(&)). From here 8 (&) 0 1+$2-1+4-2 0 1 for any ( 0 0, which contradicts to 8 (&) = 1
(# !1# )# !1!1 41 for any & $ 3. It follows that ) 01 + ( ·1+.($).($) implies ) 0 2!.($)1!.($) and thus rules out equilibrium (ii).
Equilibrium (iv) requires 92 % 0, i.e. ( $ (*!1)(*!2)2*!3 = ¡ 1
*!1 +*!21 ¢!1
¡ 1 $
*!2+ *!21 ¢!1
= *2 " 1, which is a contradiction to ( 4 *2 " 1.
Note that if ) 0 10 then ( 4 *2 " 1 is implied by ) 0 1 + ( · 1+.($).($) . This is because 1+.($).($) = ($ !1)$# !1# !1 = 1
(1+# !11 )# !1 equals 14 = 25711 in the limit & # -, and has an upper bound of 49 with & = 3. For this reason, ) 0 1 + ( · 1+.($).($) implies ( 4 49() " 1) % *2 " 1 as soon as ) $ 10.
We are left with equilibrium (i) in which both +2 and +3 are played with positive probabilities. Thus the proposition holds for "2.
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Now assume that an equilibrium in " contains only singletons. By Proposition 2 and by construction of e1+, there exists an equilibrium in "+ that contains only singletons in the support, for any 6 $ 1. This contradicts the fact that +3 is played with positive probability in "2. Similarly for sequential bidding: assume that for any +0 ! supp 1 holds that +0(* " 1) = 1 if +0(*) = 1. Proposition 2 implies that this also holds in "2, which is a contradiction: +2 is played in equilibrium with positive probability.
Proof of Proposition 4
Proof. The proof is by induction. The Þrst step follows from lemma 1 which guarantees that + (*) [() " *) 2"(+$ 11) " (] = 0$ ++ ! supp 11 in "1. Now assume that the proposition holds for "+ and consider equilibrium 1++1 in "++1. Construct e1+ as in (8): e1+(+) = 1++1(+) + 1++1(; (+$ 6 + 1)) $ ++ ! ,+% Proposition 2 ensures that e1+ is equilibrium in "+. By inductive hypothesis, for any + ! supp e1+ holds +(*) [() " *) 2"(+$ e1+) " (] = 0, for any * % 6, and hence 3!(+$ e1+) = 0.
Now let + ! supp 1++1. If + (6 + 1) = 0 then from 1++1(+) 0 0 straightforwardly follows e1+(+) 0 0 and thus + ! supp e1+, and the proposition holds.
Now assume + (6 + 1) = 1. From 1++1(+) 0 0 follows e1+(; (+$ 6 + 1)) 0 0 and therefore proposition holds in "+ for ; (+$ 6 + 1). The latter strategy coincides with + in all bids below 6 + 1. Invariance 2"(+$ 1++1) = 2"(+$ e1+) for * % 6 guarantees +(*) [() " *) 2"(+$ 1++1) " (] = 0 in "++1.
It only remains to show that the proposition holds for bid 6 + 1. The expected payo! of player ' can be written as
3!(+$ 1++1) = X+
"=1
+(*) [() " *) 2"(+$ 1++1) " (] + () " (6 + 1)) 2++1(+$ 1++1) " (%
(B-6) By Lemma A.4, 3!(+$ 1++1) = 3!(+$ e1+) = 0.
Assume () " (6 + 1)) 2++1(+$ 1++1) " ( 4 0 and consider strategy ; (+$ 6 + 1) !
,+. By Lemma A.2 holds 2"(+$ 1++1) = 2"(; (+$ 6 + 1) $ 1++1) and hence 3!(; (+$ 6 + 1) $ 1++1) =
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X+
"=1
+(*) [() " *) 2"(+$ 1++1) " (] 0 3!(+$ 1++1) $ which is a contradiction to + ! supp 1++1: player ' is better o! by dropping bid 6 + 1 from his optimal strategy.
Assume now that () " (6 + 1)) 2++1(+$ 1++1) " ( 0 0, which implies
Proof. By proposition 4, 2"= *!"- holds for A<! 7CD= +-DA-=E! + played in equilib-rium 1 with positive probability, i.e. for any + ! supp 1, such that + (*) = 1. There are no gaps in the set of bids * placed in equilibrium, i.e. if * is placed in some strategy + ! supp 1, then there exists strategy +0 ! supp 1 such that +0(* " 1) = 1 (otherwise one is strictly better o! by replacing bid * in strategy + with bid * " 1). It follows that 2" = *!"- holds for A<! *'F placed in equilibrium.
Bids above * = infn
*0:P"0
"=12" $ 1o
are never placed in equilibrium because with probability 1 some bid below or equal to * wins. Assume that strict equality holds P"
For the two harmonic series in the brackets employ the Euler equation for the sum G/ =P/
+=1 1
+ of the Þrst < summands of a harmonic series, G/( ln < + H + I/, where
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H= %577 is the Euler-Mascheroni constant, and I/ is the error term converging to zero with large <. With this in mind obtain20
" where I*!1" I*!"!1converges to zero (as a sum of two converging to zero terms) for large ) and ) " *, which yields *!1" ( 1 " =!1(% From here, * is at most 63.21% of )" 1 (the value of 1 " =!1( with ( = 1, strictly decreasing in (), hence for large ) the di!erence ) " * is also large indeed. If there is no such * that the strict equality P" to the same expression as above.
Proof of Proposition 6
By induction we can show that 1 "
"!1 probability that no rival of ' plays any strategy containing bid *.
20 The same can be obtained by approximating the series with an integral #!+1 1
(
&!!./= ( ln&!!!1&!1 , again, valid only for large enough / so that ./ = 1 can be regarded as a small enough increment.
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+, and the inequality holds for any + ! ,. By proposition 4, if 1 is equilibrium and +! supp 1 then + (6) = 1 implies 2+(+$ 1) = *!+- . Now consider a "block" strategy +0, in which +0(6) = 1, for all 6 % *. Note that only player ' can win with bids 6 % * (if any of them wins). One cannot guarantee that +0 ! supp 1, therefore 2+(+0$ 1) % *!+ -for any 6 % *: a suboptimal strategy cannot deliver a higher probability of winning than an optimal one. As a result,"
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Auc-21 In Eichberger and Vinogradov (2008) only block strategies are allowed, hence for any strategy # that contains bid / and therefore also contains bid ' $ / (as bids are placed jointly) 1$(#$ s!") = 1 implies 1!(#$ s!") = 1 (if no rival plays ', then no rival also plays / # '). The inequality in this proof thus turns into an equality.
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Manuscript available at: http://arxiv.org (arXiv:1007.4264v1 [math.PR])
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equilibrium in a LUPA in terms of bidding behavior of players.
2. In equilibrium bidders place on average more than one bid.
3. About 5% of bidders place bids above the theoretical upper bound.
4. Players seem to place lower bids more frequently than theoretically predicted.