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Notebook[{
Cell[BoxData[{
    \(<< LinearAlgebra`MatrixManipulation`\), "\[IndentingNewLine]", 
    \(\(n = 61;\)\), "\[IndentingNewLine]", 
    \( (*First\ we\ derive\ the\ matrix\ of\ coefficients\ associated\ with\ \
the\ requirement\ that\ the\ matrix\ f_B\ has\ to\ be\ a\ symmetric\ matrix . \
\ Note\ that\ this\ yields\ a\ set\ of\ n*\((n - 1)\)/
          2\ constraints*) \)}], "Input"],

Cell[BoxData[
    \(\(M11 = 
        Table[SparseArray[{\((n*\((n + i - 1)\) + i + 
                    j)\) \[Rule] \((\(-1\))\), \((n*\((n + i + j - 1)\) + 
                    i)\) \[Rule] 1}, 2*n*n], {i, 1, n - 1}, {j, 1, 
            n - i}];\)\)], "Input"],

Cell[BoxData[
    \(\(M12 = Flatten[M11];\)\)], "Input"],

Cell[BoxData[
    \(\(M1 = 
        Table[Take[M12, {2*n*n*\((j - 1)\) + 1, 2*n*n*j}], {j, 
            1, \((n*\((n - 1)\)/2)\)}];\)\)], "Input"],

Cell[BoxData[
    \( (*Next, \ 
      we\ need\ to\ define\ the\ RHS\ vector\ associated\ with\ the\ \
constraints\ arising\ from\ the\ requirement\ that\ the\ matrix\ f_B\ has\ to\
\ be\ a\ symmetric\ matrix . \ 
          When\ we\ later\ use\ the\ Linear\ Programming\ function\ in\ \
Mathematica\ we\ need\ to\ tell\ it\ which\ of\ the\ constraints\ are\ \
equalities\ and\ which\ ones\ are\ inequalities . \ 
          The\ constraints\ associated\ with\ M1\ are\ equality\ constraints \
. \ We\ must\ therefore\ pair\ each\ element\ in\ the\ RHS\ vector\ with\ a\ \
zero*) \)], "Input"],

Cell[BoxData[
    \(\(b1 = Table[{0, 0}, {j, 1, n*\((n - 1)\)/2}];\)\)], "Input"],

Cell[BoxData[
    \(\(\( (*Now\ we\ derive\ the\ matrix\ of\ coefficients\ associated\ with\
\ the\ requirement\ that\ the\ probabilities\ assigned\ by\ the\ mechanism\ \
must\ add\ up\ to\ 1. \ First, \ 
      we\ deal\ with\ "\<main diagonal\>"\ outcomes, \ 
      that\ is\ all\ type - 
        constellations\ in\ which\ the\ two\ agents'\ types\ are\ identical\
*) \)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(M2 = 
        Table[SparseArray[{\((\((i - 1)\)*n + i)\) \[Rule] 
                2, \((n*\((n + i - 1)\) + i)\) \[Rule] 1}, 2*n*n], {i, 1, 
            n}];\)\)\)], "Input"],

Cell[BoxData[
    \( (*To\ complele\ the\ description\ of\ the\ first\ set\ of\ adding - 
        up\ constraints, \ 
      we\ need\ to\ define\ the\ asssociated\ RHS\ vector . \ 
          The\ constraints\ associated\ with\ M2\ are\ equality\ constraints \
. \ We\ must\ therefore\ pair\ each\ element\ in\ the\ RHS\ vector\ with\ a\ \
zero*) \)], "Input"],

Cell[BoxData[
    \(\(b2 = Table[{1, 0}, {j, 1, n}];\)\)], "Input"],

Cell[BoxData[
    \(\(\( (*Now\ we\ derive\ the\ matrix\ of\ coefficients\ associated\ with\
\ the\ adding - up\ constraint\ for\ all\ type - 
        constellations\ in\ which\ the\ two\ agents\ types\ are\ NOT\ \
identical\ \((off - diagonal\ adding - 
              up\ constraints)\)*) \)\(\[IndentingNewLine]\)\(\
\[IndentingNewLine]\)\(M31 = 
        Table[SparseArray[{\((n*\((i - 1)\) + i + j)\) \[Rule] 
                1, \((n*\((i + j - 1)\) + i)\) \[Rule] 
                1, \((\((n*\((n + i - 1)\) + i + j)\))\) \[Rule] 1}, 
            2*n*n], {i, 1, n - 1}, {j, 1, n - i}];\)\)\)], "Input"],

Cell[BoxData[
    \(\(M32 = Flatten[M31];\)\)], "Input"],

Cell[BoxData[
    \(\(M3 = 
        Table[Take[M32, {2*n*n*\((j - 1)\) + 1, 2*n*n*j}], {j, 
            1, \((n*\((n - 1)\)/2)\)}];\)\)], "Input"],

Cell[BoxData[
    \( (*Next\ we\ complete\ the\ description\ of\ the\ non - 
        diagonal\ adding - 
        up\ constraints\ by\ specifying\ the\ associated\ RHS\ vector . \ 
            The\ constraints\ associated\ with\ M3\ are\ equality\ \
constraints . \ 
            We\ must\ therefore\ pair\ each\ element\ in\ the\ RHS\ vector\ \
with\ a\ zero*) \)], "Input"],

Cell[BoxData[
    \(\(b3 = Table[{1, 0}, {j, 1, n*\((n - 1)\)/2}];\)\)], "Input"],

Cell[BoxData[
    \(\(\( (*The\ final\ step\ in\ the\ construction\ of\ the\ matrix\ of\ \
coefficients\ associated\ with\ the\ constraints\ is\ that\ we\ account\ for\ \
the\ incentive\ compatibility\ constraints . \ 
          The\ construction\ of\ the\ coefficient\ matrix\ associated\ with\ \
the\ incentive\ compatibility\ constraints\ is\ conducted\ below\ in\ a\ \
series\ of\ steps*) \)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(M41 = 
        Table[SparseArray[{\((n*\((i - 1)\) + j)\) \[Rule] p[j]}, n*n], {i, 
            1, n}, {j, 1, n}];\)\)\)], "Input"],

Cell[BoxData[
    \(\(M42 = Flatten[M41];\)\)], "Input"],

Cell[BoxData[
    \(\(M43 = 
        Table[Take[M42, {n*n*\((j - 1)\) + 1, n*n*j}], {j, 1, 
            n*n}];\)\)], "Input"],

Cell[BoxData[
    \(v43[k_] := 
      Table[Sum[\(M43[\([i]\)]\)[\([j]\)], {i, n*\((k - 1)\) + 1, n*k}], {j, 
          1, n*n}]\)], "Input"],

Cell[BoxData[
    \(\(M44 = Table[v43[k], {k, 1, n}];\)\)], "Input"],

Cell[BoxData[
    \(MA[i_] := Table[v43[i], {n}]\)], "Input"],

Cell[BoxData[
    \(MB[i_] := Table[v43[i]*t[i], {n}]\)], "Input"],

Cell[BoxData[
    \(MAB[i_] := AppendRows[MA[i], MB[i]]\)], "Input"],

Cell[BoxData[
    \(M45[i_] := AppendRows[M44, t[i]*M44]\)], "Input"],

Cell[BoxData[
    \(MAC[i_] := MAB[i] - M45[i]\)], "Input"],

Cell[BoxData[
    \(MBC[
        i_] := {\(MAC[i]\)[\([i - 1]\)], \(MAC[i]\)[\([i + 1]\)]}\)], "Input"],

Cell[BoxData[
    \(\(MCC = Flatten[Table[MBC[i], {i, 2, n - 1}]];\)\)], "Input"],

Cell[BoxData[
    \(\(MDC = 
        Table[Take[MCC, {2*n*n \((i - 1)\) + 1, 2*n*n*i}], {i, 1, 
            2*\((n - 2)\)}];\)\)], "Input"],

Cell[BoxData[
    \(\(MEC = Prepend[MDC, \(MAC[1]\)[\([2]\)]];\)\)], "Input"],

Cell[BoxData[
    \(\(MIC = Append[MEC, \(MAC[n]\)[\([n - 1]\)]];\)\)], "Input"],

Cell[BoxData[
    \( (*Matrix\ MIC\ contains\ the\ coefficients\ associated\ with\ the\ IC\ \
constraints*) \)], "Input"],

Cell[BoxData[
    \( (*Finally, \ we\ need\ to\ create\ the\ associated\ RHS - vector, \ 
      which\ is\ simply\ a\ list\ of\ zeroes\ of\ length\ n*\((n - 1)\) . \ 
            The\ constraints\ associated\ with\ MIC\ are\ greater - or - 
        equal\ constraints . \ 
            We\ must\ therefore\ pair\ each\ element\ in\ the\ RHS\ vector\ \
with\ a\ one*) \)], "Input"],

Cell[BoxData[
    \(\(bIC = Table[{0, 1}, {2 + 2*\((n - 2)\)}];\)\)], "Input"],

Cell[BoxData[
    \( (*We\ next\ join\ the\ coefficient\ matrices\ M1, \ M2, \ 
      M3\ and\ MIC\ derived\ separately\ above\ into\ one\ big\ coefficient\ \
matrix\ for\ our\ Linear\ Programming\ Problem*) \)], "Input"],

Cell[BoxData[
    \(\(A = Normal[AppendColumns[M1, M2, M3, MIC]];\)\)], "Input"],

Cell[BoxData[
    \( (*We\ now\ join\ the\ associated\ RHS\ vectors\ \((including\ the\ \
equality\ vs . \ inequality\ dummy)\)\ b1, \ b2, \ b3\ and\ bIC, \ 
      derived\ separately\ above*) \)], "Input"],

Cell[BoxData[
    \(\(v = AppendColumns[b1, b2, b3, bIC];\)\)], "Input"],

Cell[BoxData[
    \( (*Finally, \ 
      we\ need\ the\ vector\ c\ of\ coefficients\ associated\ with\ the\ \
objective\ function\ that\ we\ are\ looking\ to\ maximize\ subject\ to\ the\ \
above\ constraints*) \)], "Input"],

Cell[BoxData[
    \( (*We\ first\ construct\ a\ matrix\ of\ coefficients\ associated\ with\ \
the\ expected\ payoff\ from\ truthful\ revelation\ for\ each\ type\ of\ agent\
\ 1. \ This\ is\ matrix\ SW1\ below*) \)], "Input"],

Cell[BoxData[
    \(\(Table[M44[\([i]\)]*t[i], {i, 1, n}];\)\)], "Input"],

Cell[BoxData[
    \(\(SW1 = 
        AppendRows[M44, Table[M44[\([i]\)]*t[i], {i, 1, n}]];\)\)], "Input"],

Cell[BoxData[
    \( (*We\ next\ produce\ a\ single\ vector\ of\ coefficients\ associated\ \
with\ the\ ex\ ante\ expected\ welfare\ of\ agent\ 1. \ The\ ex\ ante\ \
welfare\ vector\ for\ agent\ 2\ is\ the\ same\ due\ to\ symmetry\ of\ the\ \
mechanism, \ 
      so\ we\ can\ obtain\ social\ welfare\ simply\ by\ multiplying\ the\ ex\ \
ante\ welfare\ of\ agent\ 1\ by\ 2. \ Note\ that\ the\ Linear\ Programming\ \
function\ implemented\ in\ Mathematica\ ALWAYS\ looks\ for\ a\ minimum, \ 
      and\ hence\ we\ must\ multiply\ all\ the\ coefficients\ of\ the\ \
objective\ function\ by\ \((\(-1\))\)\ to\ get\ the\ right\ answer*) \)], \
"Input"],

Cell[BoxData[
    \(\(c = Sum[SW1[\([i]\)]*p[i]*\((\(-2\))\), {i, 1, n}];\)\)], "Input"],

Cell[BoxData[
    \( (*Before\ calling\ the\ LinearProgramming\ function\ in\ Mathematica\ \
we\ need\ to\ derive\ one\ last\ \(list : \ 
          for\ each\ of\ our\ 2*n*n\ unknowns\), \ 
      we\ must\ give\ lower\ and\ upper\ bounds\ for\ the\ unknowns, \ 
      which, \ in\ our\ case, \ 
      are\ probabilities\ and\ are\ hence\ between\ zero\ and\ \(\(one\)\(.\)\
\)*) \)], "Input"],

Cell[BoxData[
    \(\(l = Table[{0, 1}, {2*n*n}];\)\)], "Input"],

Cell[BoxData[
    \( (*We\ are\ now\ in\ a\ position\ to\ call\ the\ Linear\ Programming\ \
function\ implemented\ in\ Mathematica . \ For\ this\ purpose, \ 
      we\ need\ to\ assign\ numerical\ values\ to\ the\ agents\ types\ as\ \
well\ as\ the\ type\ probabilities . \ First\ we\ assign\ the\ type - 
        values, \ 
      which\ are\ evenly\ spaced\ over\ the\ unit\ interval*) \)], "Input"],

Cell[BoxData[
    \(\(t[i_] := \((i - 1)\)/\((n - 1)\);\)\)], "Input"],

Cell[BoxData[
    \( (*Before\ computing\ Second\ Mechanisms\ we\ derive\ our\ \(benchmark \
: \ the\ matrix\ "\<fbeff\>"\ of\ probabilities\ of\ implementing\ the\ \
compromise\ under\ a\ first\ best\ mechanism\)*) \)], "Input"],

Cell[BoxData[
    \(Do[{u[i_] := SparseArray[{\((n + \((1 - i)\))\) \[Rule] 1}, n]}, {i, 1, 
        n}]\)], "Input"],

Cell[BoxData[
    \(Do[{w[i_] := Table[Sum[u[j], {j, 1, \((n - \((i - 1)\))\)}]]}, {i, 1, 
        n}]\)], "Input"],

Cell[BoxData[
    \(\(fbeff = 
        Normal[Table[w[\((n - \((i - 1)\))\)], {i, 1, n}]];\)\)], "Input"],

Cell[BoxData[
    \( (*Next\ we\ compute\ the\ second\ best\ decision\ rule\ for\ the\ case\
\ where\ the\ n\ types\ have\ equal\ probability\ weight*) \)], "Input"],

Cell[BoxData[
    \(\(p[i_] := 1/n;\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(xuni = 
        LinearProgramming[c, A, v, l, \ 
          Method -> "\<InteriorPoint\>"];\)\)], "Input"],

Cell[BoxData[
    RowBox[{\(LinearProgramming::"lpipp"\), \(\(:\)\(\ \)\), "\<\"Warning: \
Method->InteriorPoint specified for non-machine precision problem. Machine \
precision result will be given. If a result of non-machine precision is \
needed, set the option to Method->\\!\\(\\\"RevisedSimplex\\\"\\). \
\\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\
\\\", ButtonFrame->None, \
ButtonData:>\\\"LinearProgramming::lpipp\\\"]\\)\"\>"}]], "Message"]
}, Open  ]],

Cell[BoxData[
    \( (*The\ following\ command\ reads\ out\ the\ second\ n*
        n\ solution\ probabilities, \ 
      which\ pertain\ to\ alternative\ B . \ 
            The\ Plot\ of\ FBuni\ illustrates\ the\ compromise\ probabilities\
\ for\ all\ possible\ type\ pairs . \ 
            Note\ that\ the\ types\ of\ agent\ 1\ are\ listed\ on\ the\ \
Vertical\ Axis\ \((in\ increasing\ order)\)\ and\ those\ of\ agent\ 2\ on\ \
the\ Horizontal\ Axis . \ 
            Black\ in\ the\ following\ Plots\ indicates\ that\ the\ \
probability\ associated\ with\ a\ type - vector\ is\ one, \ 
      while\ white\ indicates\ that\ the\ probability\ associated\ with\ a\ \
type\ pair\ is\ zero . \ 
          Grey\ indicates\ a\ probability\ between\ zero\ and\ one*) \)], \
"Input"],

Cell[BoxData[
    \(\(FBuni = 
        Table[Take[xuni, {n*n + n*\((j - 1)\) + 1, n*n + n*j}], {j, 1, 
            n}];\)\)], "Input"],

Cell[BoxData[
    \(pic3 = 
      ListDensityPlot[FBuni, ColorFunction \[Rule] \((GrayLevel[1 - #] &)\), 
        Mesh \[Rule] False, Frame \[Rule] False]\)], "Input"],

Cell[BoxData[
    \( (*The\ next\ command\ creates\ the\ cross\ diagonal\ in\ the\ unit\ \
square, \ 
      while\ pic4\ is\ the\ graphic\ object\ that\ shows\ the\ graphical\ \
representation\ of\ this\ cross\ diagonal*) \)], "Input"],

Cell[BoxData[
    \(\(diagonal = Line[{{0, 61}, {61, 0}}];\)\)], "Input"],

Cell[BoxData[
    \(\(pic4 = Show[Graphics[diagonal]];\)\)], "Input"],

Cell[BoxData[
    \( (*The\ next\ command\ merges\ the\ two\ graphic\ objects\ pic3\ and\ \
pic4\ to\ create\ Figure\ 1\ in\ the\ paper*) \)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Show[{pic3, pic4}, Frame \[Rule] True, Axes \[Rule] False, 
      FrameLabel -> {t1, t2}, 
      FrameTicks \[Rule] {{{0, 0}, {29, 0.5}, {60, 1}}, {{0, 0}, {29, 
              0.5}, {60, 1}}}]\)], "Input"],

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Cell[BoxData[
    \( (*The\ following\ command\ computes\ the\ ex\ ante\ inefficiency\ \
associated\ with\ the\ second\ best\ decision\ rule\ under\ the\ uniform\ \
type\ distribution . \ 
          We\ compute\ the\ ex\ ante\ expected\ probability\ of\ the\ \
compromise\ under\ the\ first\ best\ and\ under\ the\ second\ best\ decision\ \
rule . \ Before\ subtracting\ the\ latter\ from\ the\ former\ we\ \
"\<remove\>"\ the\ "\<main diagonal\>"\ \((i . e . \ 
              all\ type\ pairs\ which\ add\ up\ to\ one)\), \ 
      as\ it\ is\ irrelevant\ for\ efficiency\ what\ the\ probability\ of\ \
the\ compromise\ is\ for\ type\ pairs\ on\ the\ main\ diagonal*) \)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\((Sum[\(fbeff[\([i]\)]\)[\([j]\)], {i, 1, n}, {j, 1, n}]*\((1/
              n)\)*\((1/
              n)\))\) - \((Sum[\(fbeff[\([i]\)]\)[\([n - \((i - 1)\)]\)], {i, 
              1, n}]*\((1/n)\)*\((1/
              n)\))\) - \((\((Sum[\(FBuni[\([i]\)]\)[\([j]\)], {i, 1, n}, {j, 
                  1, n}]*\((1/n)\)*\((1/
                  n)\))\) - \((Sum[\(FBuni[\([i]\)]\)[\([n - \((i - 
                          1)\)]\)], {i, 1, n}]*\((1/n)\)*\((1/
                  n)\))\))\)\)], "Input"],

Cell[BoxData[
    \(0.005194663200108196`\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \( (*The\ following\ vector\ contains\ the\ n*
          n\ probabilities\ associated\ with\ alternative\ A . \ 
            The\ plot\ below\ illustrates\ them\ graphically\ for\ all\ type \
- pairs*) \)], "Input"],

Cell[BoxData[
    \(\(FBuniA = 
        Table[Take[xuni, {n*\((j - 1)\) + 1, n*j}], {j, 1, n}];\)\)], "Input"],

Cell[BoxData[
    \( (*The\ next\ command\ generates\ the\ graphics\ object\ pic5\ that\ \
represents\ the\ probabilities\ of\ alternative\ A\ under\ the\ second\ best\ \
mechanism*) \)], "Input"],

Cell[BoxData[
    \(\(pic5 = 
        ListDensityPlot[FBuniA, 
          ColorFunction \[Rule] \((GrayLevel[1 - #] &)\), Mesh \[Rule] False, 
          Frame \[Rule] False];\)\)], "Input"],

Cell[BoxData[
    \( (*The\ next\ command\ creates\ Figure\ 3\ in\ the\ paper*) \)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
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      FrameLabel -> {t1, t2}, 
      FrameTicks \[Rule] {{{0, 0}, {29, 0.5}, {60, 1}}, {{0, 0}, {29, 
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Cell[BoxData[
    \( (*Next\ we\ derive\ Figure\ 2\ in\ the\ paper . \ 
          For\ this\ purpose\ we\ define\ the\ function\ U[
          i]\ which\ is\ the\ interim\ expetced\ payoff\ of\ an\ agent\ of\ \
type\ t[i]*) \)], "Input"],

Cell[BoxData[
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          Sum[p[j]*\(FBuni[\([i]\)]\)[\([j]\)]*t[i], {j, 1, n}])\)\)], "Input"],

Cell[BoxData[
    \( (*Next\ we\ compute\ the\ interim\ expected\ utility\ for\ each\ type\ \
under\ the\ second\ best\ decision\ rule\ when\ types\ are\ all\ equally\ \
likely*) \)], "Input"],

Cell[BoxData[
    \(\(InterimU = Table[U[i], {i, 1, n}];\)\)], "Input"],

Cell[BoxData[
    \( (*The\ following\ command\ creates\ the\ graphics\ object\ pic1\ that\ \
represents\ graphically\ the\ n\ interim\ expected\ utilities, \ 
      namely\ one\ per\ type*) \)], "Input"],

Cell[BoxData[
    \(pic1 = 
      ListPlot[InterimU, PlotJoined \[Rule] False, \ 
        AxesLabel \[Rule] {ti, Ui}, \ 
        Ticks \[Rule] {{{0, 0}, {29, 0.5}, {60, 1}}, Automatic}]\)], "Input"],

Cell[BoxData[
    \( (*The\ next\ two\ commands\ generate\ a\ horizontal\ line\ at\ interim\
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Cell[BoxData[
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Cell[BoxData[
    \(\(pic2 = Show[Graphics[horizontal], Axes \[Rule] True];\)\)], "Input"],

Cell[BoxData[
    \( (*The\ next\ command\ merges\ the\ two\ graphics\ objects\ pic1\ and\ \
pic2\ to\ generate\ Figure\ 2\ in\ the\ paper*) \)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
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