25. Palacios-Huertaの発見
• 欧州トップリーグのデータから検証
– 30回以上キック/セーブした選手のみ注目
– テニスでの仮説1と仮説2はともに成立
• 実際の結果がナッシュ均衡とほぼ一致
(上が理論予測、下が実際の頻度)
2016年9月 25
their natural choices.V The typical penalty kick may then be described by the simple 2 x 2
model outlined in Section 2. Penalty kicks in this model have a unique Nash equilibrium and the
equilibrium requires each player to use a mixed strategy. As mentioned earlier, equilibrium theory
makes two testable predictions about the behaviour of kickers and goalkeepers: (1) winning
probabilities should be the same across strategies for both players, and (2) each player's strategic
choices must be serially independent.
Before we begin any formal test, it is worth examining the extent to which observed
behaviour is close to the Nash equilibrium predictions. For all players in the sample the empirical
scoring probabilities are
kL
1- kt.
where, as indicated above, kt. and gL denote the non-natural sides. The mixed strategy Nash
equilibrium predicted frequencies for these empirical values and the actual mixing probabilities
observed in the sample are
gL (%) 1-gL (%) kt. (%) 1 - kt. (%)
Nash predicted frequencies 41·99 58·01 38·54 61·46
Actual frequencies 42·31 57·69 39·98 60·02
10. As a referee noted, the assumption that the game is identical for the two kinds of kickers up to the renaming
of the actions is not obvious. Hence, we have tested this assumption using a regression framework. The null hypothesis
26. 真ん中を加えると?
• ゲームの「3要素」
– プレイヤー:キッカーとキーパー
– 戦略:左(L)か真ん中(M)か右(R)
– 利得:ゴールが決まる確率
• 真ん中は選ばれない(損する)場合も
– Chiappori, P. A., Levitt, S., & Groseclose, T. (2002).
Testing mixed-strategy equilibria when players are heterogeneous:
The case of penalty kicks in soccer. American Economic Review,
1138-1151.
2016年9月 26
27. 各マスの相対頻度は理論通り
level. Standarderrorsarein parentheses.For shotsinvolving
havebeenreversedso thatshootingleft correspondsto the "n
TABLE3-OBSERVED MATRIXOFSHOTSTAKEN
Kicker
Goalie Left Middle Right Total
Left 117 48 95 260
Middle 4 3 4 11
Right 85 28 75 188
Total 206 79 174 459
Notes: The sample includes all Frenchfirst-leaguepenalty
kicks from 1997-1999 and all Italian first-league kicks
(1997-2000). For shots involving left-footed kickers, the
directionshave been reversed so that shooting left corre-
spondsto the "natural"side for all kickers.
model is
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2016年9月 27
28. 真ん中を狙うのは吉!?
• 成功頻度が高いのは「集計の罠」かも
– ゴール確率の高いキッカーだけMを選ぶ
– 集計によってMの成功率の高さを過大評価
2016年9月 28
THEAMERICANECONOMICRE
TABLE4--OBSERVEDMATRIXOFOUTCOMES:
PERCENTAGEOFSHOTSINWHICHA GOALIS SCORED
Kicker
Goalie Left Middle Right Total
Left 63.2 81.2 89.5 76.2
Middle 100 0 100 72.7
Right 94.1 89.3 44.0 73.4
Total 76.7 81.0 70.1 74.9
Notes: The sample includes all Frenchfirst-leaguepenalty
kicks from 1997-1999 and all Italian first-league kicks
jointly in
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1148
43. 理論が外れるケースも…
• アメフト(4thダウン・ギャンブル)
– Romer, D. (2006).
Do firms maximize? Evidence from professional football. Journal of Political Economy,
114(2), 340-365.
– ギャンブルは多くの場合勝率を上げるがほとんど取られない → 勝率以外の動機?
• テニス(サーブのイン確率)
– Klaassen, F. J., & Magnus, J. R. (2009).
The efficiency of top agents: An analysis through service strategy in tennis. Journal
of Econometrics, 148(1), 72-85.
– 2本目のサーブの方が1本目よりも非効率性が高い → 最適化の失敗?
• 野球とアメフト
– Kovash, K., & Levitt, S. D. (2009).
Professionals do not play minimax: evidence from major League Baseball and the
National Football League (No. w15347). NBER
– ピッチャーはストレートを投げ過ぎ、アメフトはパスをしなさ過ぎ
2016年9月 43
46. 運に左右される延長戦
• 先攻の勝率は約60%(後攻の1.5倍!)
• 遥かに公平な決着方法がある
– 代案1:ケーキカット方式
• 攻撃開始のヤード数を提示 → 相手が攻守を決定
– 代案2:オークション方式
• 両チーム同時に攻撃権を(ヤード数で)競り合う
• 代案2の方が1よりもさらに公平
– Che, Y. K., & Hendershott, T. (2008).
How to Divide the Possession of a Football?. Economics Letters,
99(3), 561-565.
2016年9月 46