auction.nb
1
Three (and More) Player Game on the
Second Price Auction: An Example
Toshitaka Fukiharu
(Faculty of Economics, Hiroshima University)
August, 2009
Introduction
As pointed out in McMillan [2002], in 1994 the United States government adopted an auction in selling the right on the
radio frequency for pager (or beeper) service providers. Since then the governments all over the world have adopted
auctions in selling the newly invented rights for the private corporations, and the estimated revenue amounts to $120
billion. The theory of auction was first formalized by W. Vickery in 1961. In 1996, Vickery was awarded the Nobel
Memorial Prize in Economic Sciences. His classical contribution was one of the standards in the Game theory and it is
contained in standard textbooks, such as Osborne and Rubinstein [1994]. Especially, the proposition on the second
price auction is one of the standard exercises (Osborne and Rubinstein [1994, EXCERCISE 18.3, p.18]). The aim of
this paper is to actually construct payoff matrix for three player game by simulation approach, and show that the weak
dominance holds for the second price auction.
1. 2 Player Auction Game
The most common auction is the first price auction, in which the winner is the highest bidder and the winner pays the
bid price to aquire the object. W. Vickery analyzed the second price auction, in which the winner is the highest bidder,
while the winner pays the second highest bid price to aquire the object and showed that this auction has a special
feature. For the purpose of easy understanding of the Mathematica programming, we start with the 2 player auction
game and show that in the second price auction each player's bidding the true value for the player is a weakly dominant
strategy; its payoff is at least as high as the player's payoffs when the player submits any other bid, regardless
of the strategies of the other players.
auction.nb
2
ü Case1: When sa πsb
Suppose that there are 2 players, A and B. They submit bidding in a auction for an object. Their true valuations of the
object are va and vb . Thier strategy sets are S={0, 10, 20, ..., 10n} , where 0< va , vb <10n. When player A selects a
strategy, sa œS and player B selects a strategy sb œ S , and sa >sb , player A is the winner, and A pays sb to aquire the
object. Then A's payoff is va -sa and B's payoff is 0. In order to construct a fuction, check1, to compute each player's
payoff, given va , vb , sa , and sb . when sa ∫sb . Suppose that va =40, vb =20, sa =60, and sb =50. We define c1 as the
greater one in {sa , sb }, and c2 as the smaller one in {sa , sb }, where A's strategy is placed as the first element.
In[1]:=
va = 40; vb = 20; sa = 60; sb = 50;
In[2]:=
c1 = Max@sa, sbD
Out[2]=
60
In[3]:=
c2 = Min@sa, sbD
Out[3]=
50
We define c3 as the position of c1 , showing which player selects the greater one in {sa , sb }.
In[4]:=
c3 = Position@8sa, sb<, c1D@@1, 1DD
Out[4]=
1
We define pa as the payoff of the player A.
In[5]:=
pa = If@c3
Out[5]=
−10
1, va − c2, 0D
We define pb as the payoff of the player B.
In[6]:=
pb = If@c3
Out[6]=
0
2, vb − c2, 0D
Collecting these programs, we define a function, check1[va , vb , sa , sb ], which computes { pa , pb } given va , vb , sa , and
sb when sa ∫sb .
In[7]:=
Clear@va, vb, sa, sb, c1, c2, c3, pa, pbD
In[8]:=
check1@va_, vb_, sa_, sb_D := Module@8c1, c2, c3, pa, pb<,
c1 = Max@sa, sbD; c2 = Min@sa, sbD; c3 = Position@8sa, sb<, c1D@@1, 1DD;
pa = If@c3 1, va − c2, 0D; pb = If@c3 2, vb − c2, 0D; 8pa, pb<D
In[9]:=
check1@40, 20, 60, 50D
Out[9]=
8−10, 0<
auction.nb
3
ü Case2: When sa =sb
When player A selects a strategy, sa œS and player B selects a strategy sb œ S , and sa =sb , we cannot compute the
payoffs for the both players by the function, check1. When sa =sb the winner is selected by lotttery, so that the probability for each player becomes the winner ia 1/2. If the winner is player A, A pays sa =sb to aquire the object, and the
payoff of player A is the expected value, (va -sa )/2. If the winner is player B, B pays sa =sb to aquire the object, and the
payoff of player B is the expected value, (vb -sb )/2. Suppose that va =20, vb =40, sa =10, and sb =10. The payoff for A,
pa =5, and the one for B , pb =15.
In[10]:=
va = 20; vb = 40; sa = 10; sb = 10;
In[11]:=
pa = Hva − saL ê 2
Out[11]=
5
In[12]:=
pb = Hvb − sbL ê 2
Out[12]=
15
The function, check2[va , vb , sa , sb ] computes { pa , pb } given va , vb , sa , and sb when sa =sb .
In[13]:=
Clear@va, vb, sa, sb, c1, c2, c3, pa, pbD
In[14]:=
check2@va_, vb_, sa_, sb_D :=
Module@8c1, c2, c3, pa, pb<, pa = Hva − saL ê 2; pb = Hvb − sbL ê 2; 8pa, pb<D
In[15]:=
check2@20, 40, 10, 10D
Out[15]=
85, 15<
ü General Case
In general, assembling the functions check1 and check2, check3[va , vb , sa , sb ] is constructed, which computes
{ pa , pb } when va , vb , sa , and sb are given arbitrarily.
In[16]:=
check3@va_, vb_, sa_, sb_D := Which@sa 0 && sb 0, 80, 0<,
sa sb, check2@va, vb, sa, sbD, sa =!= sb, check1@va, vb, sa, sbDD
In[17]:=
check3@20, 40, 10, 10D
Out[17]=
85, 15<
ü Payoff Matrix When va =50, vb =30，sa , sb =0～70
When va =50, vb =30 and strategy set of bidding prices is S, the payoff matrix of this auction game is constructed in
what follows. On the one hand, the collection of payoffs for player A with strategy sa =50 is {50, 40, 30, 20, 10, 0, 0,
0}, which is better than any other collection of payoffs for player A with strategy sa ∫50. On the other hand, the
collection of payoffs for player B with strategy sb =30 is {30, 20, 10, 0, 0, 0, 0, 0}, which is better than any other
collection of payoffs for player A with strategy sb ∫30. Thus, {50, 30} is weakly dominant set of strategies.
auction.nb
In[18]:=
4
Hcheck4 = Table@check3@50, 30, 10 ∗ i, 10 ∗ jD, 8i, 0, 7<, 8j, 0, 7<DL êê MatrixForm
0
i
j
J N
j
j
0
j
j
j
j
j
j
50
j
j
J
N
j
j
j
j 0
j
j
j
j
50
j
j
j
J
N
j
j
0
j
j
j
j
j
50
j
j
j
J
N
j
j
0
j
j
j
j
j
50
j
j
j
J
N
j
j
j 0
j
j
j
j
j 50
j
j
J
N
j
j
j
0
j
j
j
j
j
50
j
j
j
J
N
j
j
0
j
j
j
j
j
j 50
j
jJ
N
k 0
J
0
N
30
20
J
N
10
40
J
N
0
40
J
N
0
40
J
N
0
40
J
N
0
40
J
N
0
40
J
N
0
Out[18]//MatrixForm=
J
0
N
30
0
J
N
20
15
J
N
5
30
J
N
0
30
J
N
0
30
J
N
0
30
J
N
0
30
J
N
0
J
0
N
30
0
J
N
20
0
J
N
10
10
J
N
0
20
J
N
0
20
J
N
0
20
J
N
0
20
J
N
0
0
0
0
N J
N
J
N
30
30
30
0
0
0
J
N J
N
J
N
20
20
20
0
0
0
J
N J
N
J
N
10
10
10
0
0
0
J N
J N
J N
0
0
0
5
0
0
J
N J
N J
N
−5
−10
−10
10
0
0
J
N J
N J
N
0
−10
−20
−5
10
0
J
N
J N J
N
0
0
−15
10
0
−10
J
N
J N J
N
0
0
0
J
0
z
N y
z
30 z
z
z
z
z
z
z
0
z
z
J
N z
z
20 z
z
z
z
z
z
0
z
z
J
N z
z
10 z
z
z
z
z
z
0
z
z
J N z
z
z
0
z
z
z
z
z
0
z
z
J
Nz
z
z
−10 z
z
z
z
z
z
0
z
J
Nz
z
z
−20 z
z
z
z
z
z
0
z
z
J
Nz
z
−30 z
z
z
z
z
−10 z
z
z
J
Nz
−20 {
J
2. 3 Player Auction Game
In this section, it is shown that the weak dominance holds for 3 player auction game, by constructing concrete payoffs
for 3 players. Compared with the 2 player game, somewhat complicated procedure is required. We must consider 3
cases. The 1st case is the one in which the only one among the 3 bidding prices is the greatest, the 2nd is the one in
which only two among the 3 bidding prices are the same greatest, while the 3rd is the one in which all the bidding
prices are the same.
ü Case1: When sa >sb , sc
Consider the 1st case, in which the only one among the 3 bidding prices is the greatest. Suppose that va =50, vb =40,
vb =30, sa =40, sb =30, sb =30. It is ascertained that the only one among the 3 bidding prices is the greatest.
In[19]:=
va = 50; vb = 40; vc = 30; sa = 40; sb = 30; sc = 30;
In[20]:=
Length@Flatten@Position@8sa, sb, sc<, Max@8sa, sb, sc<DDDD
Out[20]=
1
We define c1 as the greatest element in {sa , sb , sc }.
In[21]:=
c1 = Max@sa, sb, scD
Out[21]=
40
We define c2 as the second greatest element in {sa , sb , sc }.
In[22]:=
c2 = Reverse@Sort@8sa, sb, sc<DD@@2DD
Out[22]=
30
We define c3 as the position of c1 in {sa , sb , sc }.
auction.nb
5
In[23]:=
c3 = Position@8sa, sb, sc<, c1D@@1, 1DD
Out[23]=
1
We define pa as the payoff of the player A.
In[24]:=
pa = If@c3
Out[24]=
20
1, va − c2, 0D
We define pb as the payoff of the player B.
In[25]:=
pb = If@c3
Out[25]=
0
2, vb − c2, 0D
We define pc as the payoff of the player C.
In[26]:=
pc = If@c3
Out[26]=
0
3, vc − c2, 0D
Collecting these programs, we define a function, check1[va , vb , vc , sa , sb , sc ], which computes { pa , pb , pc } given va ,
vb , vc , sa , sb , and sc for the case 1.
In[27]:=
Clear@check1, va, vb, vc, sa, sb, sc, c1, c2, c3, pa, pb, pcD
In[28]:=
check1@va_, vb_, vc_, sa_, sb_, sc_D := Module@8c1, c2, c3, pa, pb, pc<,
c1 = Max@sa, sb, scD; c2 = Reverse@Sort@8sa, sb, sc<DD@@2DD;
c3 = Position@8sa, sb, sc<, c1D@@1, 1DD; pa = If@c3 1, va − c2, 0D;
pb = If@c3 2, vb − c2, 0D; pc = If@c3 3, vc − c2, 0D; 8pa, pb, pc<D
In[29]:=
check1@50, 40, 30, 40, 30, 30D
Out[29]=
820, 0, 0<
ü Case 2: When sa =sb >sc .
Consider the 2nd case, in which only two among the 3 bidding prices are the same greatest. Suppose that va =50,
vb =40, vb =30, sa =40, sb =40, sb =20. It is ascertained that the only two among the 3 bidding prices are the same
greatest.
In[30]:=
va = 50; vb = 40; vc = 30; sa = 40; sb = 40; sc = 20;
In[31]:=
Length@Flatten@Position@8sa, sb, sc<, Max@8sa, sb, sc<DDDD
Out[31]=
2
We define c1 as the greatest element in {sa , sb , sc }.
In[32]:=
c1 = Max@sa, sb, scD
Out[32]=
40
We define c2 as the smallest element in {sa , sb , sc }, which is actually the smallest.
auction.nb
6
In[33]:=
c2 = Sort@8sa, sb, sc<D@@1DD
Out[33]=
20
We define c3 as the set of positions of c1 in {sa , sb , sc }.
In[34]:=
Out[34]=
c3 = Flatten@Position@8sa, sb, sc<, c1DD
81, 2<
We define pa as the payoff of the player A. Note that in this case, the second greatest bidding price is c1 .
In[35]:=
pa = If@Length@Intersection@81<, c3DD
Out[35]=
5
1, Hva − c1L ê 2, 0D
We define pb as the payoff of the player B.
In[36]:=
pb = If@Length@Intersection@82<, c3DD
Out[36]=
0
1, Hvb − c1L ê 2, 0D
We define pc as the payoff of the player C.
In[37]:=
pc = If@Length@Intersection@83<, c3DD
Out[37]=
0
1, Hvc − c1L ê 2, 0D
Collecting these programs, we define a function, check2[va , vb , vc , sa , sb , sc ], which computes { pa , pb , pc } given va ,
vb , vc , sa , sb , and sc for the case 2.
In[38]:=
Clear@check2, va, vb, vc, vd, sa, sb, sc, sd, c1, c2, c3, pa, pb, pc, pdD
In[39]:=
check2@va_, vb_, vc_, sa_, sb_, sc_D := Module@8c1, c3, pa, pb, pc<,
c1 = Max@sa, sb, scD; c3 = Flatten@Position@8sa, sb, sc<, c1DD;
pa = If@Length@Intersection@81<, c3DD 1, Hva − c1L ê 2, 0D;
pb = If@Length@Intersection@82<, c3DD 1, Hvb − c1L ê 2, 0D;
pc = If@Length@Intersection@83<, c3DD 1, Hvc − c1L ê 2, 0D; 8pa, pb, pc<D
In[40]:=
check2@50, 40, 30, 40, 40, 20D
Out[40]=
85, 0, 0<
ü Case 3: When sa =sb =sc .
Consider the 3rd case, in which the 3 bidding prices are the same. Suppose that va =50, vb =40, vb =30, sa =40, sb =40,
sb =40.
In[41]:=
va = 50; vb = 40; vc = 30; sa = 40; sb = 40; sc = 40;
It is ascertained thatthe the 3 bidding prices are the same greatest.
In[42]:=
Length@Flatten@Position@8sa, sb, sc<, Max@8sa, sb, sc<DDDD
Out[42]=
3
auction.nb
7
We define pa as the payoff of the player A.
In[43]:=
Out[43]=
pa = Hva − saL ê 3
10
3
We define pb as the payoff of the player B.
In[44]:=
pb = Hvb − sbL ê 3
Out[44]=
0
We define pc as the payoff of the player C.
In[45]:=
pc = Hvc − scL ê 3
Out[45]=
−
10
3
Collecting these programs, we define a function, check2[va , vb , vc , sa , sb , sc ], which computes { pa , pb , pc } given va ,
vb , vc , sa , sb , and sc for the case 3.
In[46]:=
Clear@check3, va, vb, vc, sa, sb, sc, c1, c2, c3, pa, pb, pcD
In[47]:=
check3@va_, vb_, vc_, sa_, sb_, sc_D :=
Module@8pa, pb, pc<, pa = Hva − saL ê 3; pb = Hvb − sbL ê 3; pc = Hvc − scL ê 3; 8pa, pb, pc<D
In[48]:=
check3@50, 40, 30, 40, 40, 40D
Out[48]=
9
10
10
, 0, −
=
3
3
ü General Case
In general, assembling the functions check1 and check2, and check3, check4[va , vb , vc , sa , sb , sc ] is constructed,
which computes { pa , pb , pc } when va , vb , vc , sa , sb , and sc are given arbitrarily.
In[49]:=
Clear@check4D
In[50]:=
check4@va_, vb_, vc_, sa_, sb_, sc_D := Which@sa 0 && sb 0 && sc 0,
80, 0, 0<, Length@Flatten@Position@8sa, sb, sc<, Max@8sa, sb, sc<DDDD
check3@va, vb, vc, sa, sb, scD,
Length@Flatten@Position@8sa, sb, sc<, Max@8sa, sb, sc<DDDD 2,
check2@va, vb, vc, sa, sb, scD,
Length@Flatten@Position@8sa, sb, sc<, Max@8sa, sb, sc<DDDD 1,
check1@va, vb, vc, sa, sb, scDD
In[51]:=
Out[51]=
check4@60, 40, 30, 10, 10, 20D
80, 0, 20<
3,
auction.nb
8
ü Payoff Matrix When va =50, vb =40, vc =30, and sa , sb , sc =0～70
When va =50, vb =40, and vc =30, and the strategy set of bidding prices is S, the payoff matrix of this auction game is
constructed in what follows. First of all, the collection of payoffs for player A with strategy sa =50 is {50, 40, 30, 20,
10, 0, 0, 0, 40, 40, 30, 20, 10, 0, 0, 0, 30, 30, 30, 20, 10, 0, 0, 0, 20, 20, 20, 20, 10, 0, 0, 0, 10, 10, 10, 10, 10, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, which is shown to be better than any other collection of
payoffs for player A with strategy sa ∫50. Second, the collection of payoffs for player B with strategy sb =40 is {40, 30,
20, 10, 0, 0, 0, 0, 30, 30, 20, 10, 0, 0, 0, 0, 20, 20, 20, 10, 0, 0, 0, 0, 10, 10, 10, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, which is shown to be better than any other collection of
payoffs for player B with strategy sb ∫40. Finally, the collection of payoffs for player C with strategy sc =30 is {30, 20,
10, 0, 0, 0, 0, 0, 20, 20, 10, 0, 0, 0, 0, 0, 10, 10, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, which is shown to be better than any other collection of payoffs for
player C with strategy sc ∫30. Thus, {50, 40, 30} is shown to be weakly dominant set of strategies.
In[52]:=
va0 = 50; vb0 = 40; vc0 = 30;
ü Comparison between sa =50 and sa π50
The collection of payoffs for player A with strategy sa =50, A5 is constructed as in what follows.
In[53]:=
In[54]:=
Out[54]=
ga@i0_D := Module@8m1, m2<, Hm1 = Table@
check4@va0, vb0, vc0, 10 ∗ i, 10 ∗ j, 10 ∗ kD, 8i, i0, i0<, 8j, 0, 7<, 8k, 0, 7<DL;
m2 = Partition@Flatten@m1D, 3D; Table@m2@@i, 1DD, 8i, 1, Length@m2D<DD
a5 = ga@5D
850, 40, 30, 20, 10, 0, 0, 0, 40, 40, 30, 20, 10, 0, 0, 0, 30, 30, 30,
20, 10, 0, 0, 0, 20, 20, 20, 20, 10, 0, 0, 0, 10, 10, 10, 10, 10, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0<
The collection of payoffs for player A with strategy sa =10, A1 , is constructed, and A5 -A1 is computed as in what
follows. A5 ¥A1 and A5 ∫A1 .
In[55]:=
Out[55]=
a1 = ga@1D; a5 − a1
90, 20, 30, 20, 10, 0, 0, 0, 20,
80
, 30, 20, 10, 0, 0, 0, 30, 30, 30,
3
20, 10, 0, 0, 0, 20, 20, 20, 20, 10, 0, 0, 0, 10, 10, 10, 10, 10, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0=
The collection of payoffs for player A with strategy sa =20, A2 , is constructed, and A5 -A2 is computed as in what
follows. A5 ¥A2 and A5 ∫A2 .
In[56]:=
Out[56]=
a2 = ga@2D; a5 − a2
80, 0, 15, 20, 10, 0, 0, 0, 0, 0, 15, 20, 10, 0, 0, 0, 15, 15, 20,
20, 10, 0, 0, 0, 20, 20, 20, 20, 10, 0, 0, 0, 10, 10, 10, 10, 10, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0<
The collection of payoffs for player A with strategy sa =30, A3 , is constructed, and A5 -A3 is computed as in what
follows. A5 ¥A3 and A5 ∫A3 .
auction.nb
In[57]:=
Out[57]=
9
a3 = ga@3D; a5 − a3
90, 0, 0, 10, 10, 0, 0, 0, 0, 0, 0, 10, 10, 0, 0, 0, 0, 0, 0, 10,
40
10, 0, 0, 0, 10, 10, 10,
, 10, 0, 0, 0, 10, 10, 10, 10, 10, 0, 0, 0,
3
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0=
The collection of payoffs for player A with strategy sa =40, A4 , is constructed, and A5 -A4 is computed as in what
follows. A5 ¥A4 and A5 ∫A4 .
In[58]:=
Out[58]=
a4 = ga@4D; a5 − a4
90, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0,
20
0, 5, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 5, 5, 5, 5,
, 0, 0, 0, 0,
3
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0=
The collection of payoffs for player A with strategy sa =60, A6 , is constructed, and A5 -A6 is computed as in what
follows. A5 ¥A6 and A5 ∫A6 .
In[59]:=
Out[59]=
a6 = ga@6D; a5 − a6
90, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0,
0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0,
10
0, 0, 0, 0, 5, 0, 5, 5, 5, 5, 5, 5,
, 0, 0, 0, 0, 0, 0, 0, 0, 0=
3
The collection of payoffs for player A with strategy sa =70, A7 , is constructed, and A5 -A7 is computed as in what
follows. A5 ¥A7 and A5 ∫A7 .
In[60]:=
Out[60]=
a7 = ga@7D; a5 − a7
90, 0, 0, 0, 0, 0, 10, 10, 0, 0, 0, 0, 0, 0, 10, 10, 0, 0, 0, 0, 0, 0, 10,
10, 0, 0, 0, 0, 0, 0, 10, 10, 0, 0, 0, 0, 0, 0, 10, 10, 0, 0, 0, 0, 0, 0,
20
10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10,
=
3
Thus, the collection of payoffs for player A with strategy sa =50 is {50, 40, 30, 20, 10, 0, 0, 0, 40, 40, 30, 20, 10, 0, 0,
0, 30, 30, 30, 20, 10, 0, 0, 0, 20, 20, 20, 20, 10, 0, 0, 0, 10, 10, 10, 10, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0}, which is better than any other collection of payoffs for player A with strategy sa ∫50.
ü Comparison between sb =40 and sb π40
The collection of payoffs for player B with strategy sb =40, B4 is constructed as in what follows.
In[61]:=
In[62]:=
Out[62]=
gb@j0_D := Module@8m1, m2<, Hm1 = Table@
check4@va0, vb0, vc0, 10 ∗ i, 10 ∗ j, 10 ∗ kD, 8i, 0, 7<, 8j, j0, j0<, 8k, 0, 7<DL;
m2 = Partition@Flatten@m1D, 3D; Table@m2@@i, 2DD, 8i, 1, Length@m2D<DD
b4 = gb@4D
840, 30, 20, 10, 0, 0, 0, 0, 30, 30, 20, 10, 0, 0, 0, 0, 20, 20,
20, 10, 0, 0, 0, 0, 10, 10, 10, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0<
auction.nb
10
The collection of payoffs for player B with strategy sb =10, B1 , is constructed, and B4 -B1 is computed as in what
follows. B4 ¥B1 and B4 ∫B1 .
In[63]:=
Out[63]=
b1 = gb@1D; b4 − b1
80, 15, 20, 10, 0, 0, 0, 0, 15, 20, 20, 10, 0, 0, 0, 0, 20, 20,
20, 10, 0, 0, 0, 0, 10, 10, 10, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0<
The collection of payoffs for player B with strategy sb =20, B2 , is constructed, and B4 -B2 is computed as in what
follows. B4 ¥B2 and B4 ∫B2 .
In[64]:=
Out[64]=
b2 = gb@2D; b4 − b2
90, 0, 10, 10, 0, 0, 0, 0, 0, 0, 10, 10, 0, 0, 0, 0, 10, 10,
40
, 10, 0, 0, 0, 0, 10, 10, 10, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
3
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0=
The collection of payoffs for player B with strategy sb =30, B3 , is constructed, and B4 -B3 is computed as in what
follows. B4 ¥B3 and B4 ∫B3 .
In[65]:=
Out[65]=
b3 = gb@3D; b4 − b3
90, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0,
20
5, 0, 0, 0, 0, 5, 5, 5,
, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
3
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0=
The collection of payoffs for player B with strategy sb =50, B5 , is constructed, and B4 -B5 is computed as in what
follows. B4 ¥B5 and B4 ∫B5 .
In[66]:=
Out[66]=
b5 = gb@5D; b4 − b5
90, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0,
0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 5, 5,
10
5, 5, 5,
, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0=
3
The collection of payoffs for player B with strategy sb =60, B6 , is constructed, and B4 -B6 is computed as in what
follows. B4 ¥B6 and B4 ∫B6 .
In[67]:=
Out[67]=
b6 = gb@6D; b4 − b6
90, 0, 0, 0, 0, 10, 10, 0, 0, 0, 0, 0, 0, 10, 10, 0, 0, 0, 0, 0, 0,
10, 10, 0, 0, 0, 0, 0, 0, 10, 10, 0, 0, 0, 0, 0, 0, 10, 10, 0, 10, 10, 10,
20
10, 10, 10, 10, 0, 10, 10, 10, 10, 10, 10,
, 0, 0, 0, 0, 0, 0, 0, 0, 0=
3
The collection of payoffs for player B with strategy sb =70, B7 , is constructed, and B4 -B7 is computed as in what
follows. B4 ¥B7 and B4 ∫B7 .
In[68]:=
Out[68]=
b7 = gb@7D; b4 − b7
80, 0, 0, 0, 0, 10, 20, 15, 0, 0, 0, 0, 0, 10, 20, 15, 0, 0, 0, 0, 0, 10, 20,
15, 0, 0, 0, 0, 0, 10, 20, 15, 0, 0, 0, 0, 0, 10, 20, 15, 10, 10, 10, 10, 10,
10, 20, 15, 20, 20, 20, 20, 20, 20, 20, 15, 15, 15, 15, 15, 15, 15, 15, 10<
auction.nb
11
The collection of payoffs for player B with strategy sb =40 is {40, 30, 20, 10, 0, 0, 0, 0, 30, 30, 20, 10, 0, 0, 0, 0, 20, 20,
20, 10, 0, 0, 0, 0, 10, 10, 10, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0}, which is better than any other collection of payoffs for player B with strategy sb ∫40.
ü Comparison between sc =30 and sc π30
The collection of payoffs for player C with strategy sc =30, C3 is constructed as in what follows.
In[69]:=
In[70]:=
Out[70]=
gc@k0_D := Module@8m1, m2<, Hm1 = Table@
check4@va0, vb0, vc0, 10 ∗ i, 10 ∗ j, 10 ∗ kD, 8i, 0, 7<, 8j, 0, 7<, 8k, k0, k0<DL;
m2 = Partition@Flatten@m1D, 3D; Table@m2@@i, 3DD, 8i, 1, Length@m2D<DD
c3 = gc@3D
830, 20, 10, 0, 0, 0, 0, 0, 20, 20, 10, 0, 0, 0, 0, 0, 10, 10,
10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0<
The collection of payoffs for player C with strategy sc =10, C1 , is constructed, and C3 -C1 is computed as in what
follows. C3 ¥C1 and C3 ∫C1 .
In[71]:=
Out[71]=
c1 = gc@1D; c3 − c1
90, 10, 10, 0, 0, 0, 0, 0, 10,
40
, 10, 0, 0, 0, 0, 0, 10, 10,
3
10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0=
The collection of payoffs for player C with strategy sc =20, C2 , is constructed, and C3 -C2 is computed as in what
follows. C3 ¥C2 and C3 ∫C2 .
In[72]:=
Out[72]=
c2 = gc@2D; c3 − c2
90, 0, 5, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 5, 5,
20
,
3
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0=
The collection of payoffs for player C with strategy sc =40, C4 , is constructed, and C3 -C4 is computed as in what
follows. C3 ¥C4 and C3 ∫C4 .
In[73]:=
Out[73]=
c4 = gc@4D; c3 − c4
90, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0,
10
0, 5, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 5, 5, 5, 5,
, 0, 0, 0, 0,
3
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0=
The collection of payoffs for player C with strategy sc =50, C5 , is constructed, and C3 -C5 is computed as in what
follows. C3 ¥C5 and C3 ∫C5 .
auction.nb
In[74]:=
Out[74]=
12
c5 = gc@5D; c3 − c5
90, 0, 0, 0, 10, 10, 0, 0, 0, 0, 0, 0, 10, 10, 0, 0, 0, 0, 0, 0, 10,
10, 0, 0, 0, 0, 0, 0, 10, 10, 0, 0, 10, 10, 10, 10, 10, 10, 0, 0, 10, 10,
20
10, 10, 10,
, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0=
3
The collection of payoffs for player C with strategy sc =60, C6 , is constructed, and C3 -C6 is computed as in what
follows. C3 ¥C6 and C3 ∫C6 .
In[75]:=
Out[75]=
c6 = gc@6D; c3 − c6
80, 0, 0, 0, 10, 20, 15, 0, 0, 0, 0, 0, 10, 20, 15, 0, 0, 0, 0, 0, 10, 20,
15, 0, 0, 0, 0, 0, 10, 20, 15, 0, 10, 10, 10, 10, 10, 20, 15, 0, 20, 20, 20,
20, 20, 20, 15, 0, 15, 15, 15, 15, 15, 15, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0<
The collection of payoffs for player C with strategy sc =70, C7 , is constructed, and C3 -C7 is computed as in what
follows. C3 ¥C7 and C3 ∫C7 .
In[76]:=
Out[76]=
c7 = gc@7D; c3 − c7
90, 0, 0, 0, 10, 20, 30, 20, 0, 0, 0, 0, 10, 20, 30, 20, 0, 0, 0, 0, 10, 20, 30, 20,
0, 0, 0, 0, 10, 20, 30, 20, 10, 10, 10, 10, 10, 20, 30, 20, 20, 20, 20, 20, 20,
40
20, 30, 20, 30, 30, 30, 30, 30, 30, 30, 20, 20, 20, 20, 20, 20, 20, 20,
=
3
The collection of payoffs for player C with strategy sc =30 is {30, 20, 10, 0, 0, 0, 0, 0, 20, 20, 10, 0, 0, 0, 0, 0, 10, 10,
10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
which is better than any other collection of payoffs for player C with strategy sc ∫30. Thus, {50, 40, 30} is weakly
dominant set of strategies.
Conclusions
The aim of this paper was to construct a concrete example of the weak dominance property of the second price auction,
first noticed by W. Vickery. In 1996, he was awarded the Nobel Memorial Prize in Economic Sciences. As is well
known, the most common auction is the first price auction, in which the winner is the highest bidder and the winner
pays the bid price to acquire the object. W. Vickery analyzed the second price auction, in which the winner is the
highest bidder, while the winner pays the second highest bid price to acquire the object and showed that this auction
has a special feature. For the purpose of easy understanding of the Mathematica programming, we started with the 2
player auction game and constructed an example, in which in the second price auction each player's bidding the true
value for the player is a weakly dominant strategy; its payoff is at least as high as the player's payoffs when the
player submits any other bid, regardless of the strategies of the other players. We then proceeded to the
construction of an example with 3 players, in which in the second price auction each player's bidding the true value
for the player is a weakly dominant strategy; its payoff is at least as high as the player's payoffs when the player
submits any other bid, regardless of the strategies of the other players. The extension of our programming to
the 4 player auction game, and further, is quite easy.
References
(1) McMillan, J.[2002], Reinventing the Bazaar: A Natural History of Markets, New York, W.W. Norton.
auction.nb
13
(2) Osborne, M. J. and A. Rubinstein [1994], A Course in Game Theory, Cambridge, Massachusetts, The MIT Press.
(3) Vickrey, W. [1961], "Counterspeculation, Auctions, and Competitive Sealed Tenders", Journal of Finance 16, 8-37.