A Game-Theoretic Approach to Personnel Decisions in American ...
A Game-Theoretic Approach to Personnel Decisions in American ... A Game-Theoretic Approach to Personnel Decisions in American ...
and the model is readily adaptable to accommodate this possibility, we will assume for simplicity that δ ∈ [0, 1]. In our simulations, we generally suppose that δ = 1, although we un- 2 dertake an examination of our results for a number of values of δ in Section 6. It is important to mention that each quarterback corresponds to exactly one x rating. While this rating may increase or decrease over the course of a player’s career, at any given moment it is fixed. 4 Solving the Model We solve the improvement in passing model analytically and determine Nash equilibria for the proportion of run (and pass) plays depending on x,δ and the values in Table 1. First, assume that both the offense and defense are playing optimally. Let pr be the probability that the offense calls a running play. The Nash equilibrium value for the proportion of running plays will arise when the defensive team is strategy-indifferent. That is, a Nash equilibrium occurs when the expected value of the “Defend Run” and “Defend Pass” strategies are equal under the assumption that on each play the offensive coach elects to run the ball independently with probability pr. Then R(δ,x), the Nash equilibrium value for the proportion of running plays, is the value of pr that solves the following equation: ar,r(1 + δx)pr + ap,r(1 + x)(1 − pr) = ar,p(1 + δx)pr + (1 + x)ap,p(1 − pr). (1) After some elementary algebra, we obtain R(δ,x) = (ap,r − ap,p)(1 + x) . (2) (ar,p − ar,r)(1 + δx) + (ap,r − ap,p)(1 + x) Observe that R(δ,x) depends on the quantities (ar,p − ar,r) and (ap,r − ap,p). These terms measure the benefit of a running or passing play, respectively, carried out against the “wrong” defense. For instance, if (ar,p − ar,r) is large relative to (ap,r − ap,p), then there is more added benefit for a team to run against a pass defense than it is to pass against a run defense (relative to each play being carried out against an appropriate defense). Interestingly, and somewhat counterintuitively, this bonus for a team to “guess right” with a running play results in a smaller overall proportion of running plays being called.
As a notational convenience, let D(δ,x) = (ar,p − ar,r)(1 + δx) + (ap,r − ap,p)(1 + x). To find the defensive Nash equilibrium functions, we perform a similar calculation to that used to obtain (2). The defensive Nash equilibrium functions are Rdef(δ,x) = (ar,p and − ap,p) + x(δar,p − ap,p) D (3) Pdef(δ,x) = 1 − Rdef(δ,x) (4) where Rdef(δ,x) and Pdef(δ,x) denote the Nash equilibrium proportions for run defense and pass defense, respectively. These equilibria provide optimal mixed strategy solutions to the game in terms of x and δ. We limit x to the closed interval [0, 1], because large values of x result in unreasonable increases in the expected yards per play, even for reasonable choices of ar,r, ar,p, ap,r, and ap,p. We feel that the assumption that a new quarterback will at most double the passing efficiency of a team is reasonable, and again the model is readily adaptable. Next, note that since (1+x) > (1+δx), there may exist some x0 ∈ (0, 1] such that the passing row in Table 2 will be dominant for all x ≥ x0. We refer to this value as the breaking point of the model, since it is the point where the offense would logically transition to a pure strategy. We note that breaking points do not always exist in (0, 1], and in these cases, the game is a mixed strategy game for all x ∈ (0, 1]. In fact, we put forth that the presence of a breaking point in our model provides a reasonable bound on the value of x, as it does not seem realistic that new player personnel would lead a team to completely abandon the running game, even if this may be called for in a narrow set of game situations. However, in the case that a breaking point does exist in (0, 1], the following development is necessary for completeness. Recall that we assume ar,r < ap,r and ar,p > ap,p. Under these assumptions, the breaking point is easily computed as the value of x satisfying the equation ap,p(1 + x) = ar,p(1 + δx). The following proposition reflects this. Proposition 1 If ar,r < ap,r and ar,p > ap,p, then the breaking point of the game given in Table 2 is (ar,p − ap,p) (ap,p − δar,p) .
- Page 1 and 2: 1 Introduction There is a growing i
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- Page 7 and 8: ∂R∗ ∂x = (ar,p − ar,r)(ap,r
- Page 9 and 10: in order to produce our desired pay
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and the model is readily adaptable <strong>to</strong> accommodate this possibility, we will<br />
assume for simplicity that δ ∈ [0, 1].<br />
In our simulations, we generally suppose that δ = 1,<br />
although we un-<br />
2<br />
dertake an exam<strong>in</strong>ation of our results for a number of values of δ <strong>in</strong> Section<br />
6. It is important <strong>to</strong> mention that each quarterback corresponds <strong>to</strong> exactly<br />
one x rat<strong>in</strong>g. While this rat<strong>in</strong>g may <strong>in</strong>crease or decrease over the course of a<br />
player’s career, at any given moment it is fixed.<br />
4 Solv<strong>in</strong>g the Model<br />
We solve the improvement <strong>in</strong> pass<strong>in</strong>g model analytically and determ<strong>in</strong>e Nash<br />
equilibria for the proportion of run (and pass) plays depend<strong>in</strong>g on x,δ and the<br />
values <strong>in</strong> Table 1.<br />
First, assume that both the offense and defense are play<strong>in</strong>g optimally.<br />
Let pr be the probability that the offense calls a runn<strong>in</strong>g play. The Nash equilibrium<br />
value for the proportion of runn<strong>in</strong>g plays will arise when the defensive<br />
team is strategy-<strong>in</strong>different. That is, a Nash equilibrium occurs when the<br />
expected value of the “Defend Run” and “Defend Pass” strategies are equal<br />
under the assumption that on each play the offensive coach elects <strong>to</strong> run the<br />
ball <strong>in</strong>dependently with probability pr. Then R(δ,x), the Nash equilibrium<br />
value for the proportion of runn<strong>in</strong>g plays, is the value of pr that solves the<br />
follow<strong>in</strong>g equation:<br />
ar,r(1 + δx)pr + ap,r(1 + x)(1 − pr) = ar,p(1 + δx)pr + (1 + x)ap,p(1 − pr). (1)<br />
After some elementary algebra, we obta<strong>in</strong><br />
R(δ,x) =<br />
(ap,r − ap,p)(1 + x)<br />
. (2)<br />
(ar,p − ar,r)(1 + δx) + (ap,r − ap,p)(1 + x)<br />
Observe that R(δ,x) depends on the quantities (ar,p − ar,r) and (ap,r −<br />
ap,p). These terms measure the benefit of a runn<strong>in</strong>g or pass<strong>in</strong>g play, respectively,<br />
carried out aga<strong>in</strong>st the “wrong” defense. For <strong>in</strong>stance, if (ar,p − ar,r)<br />
is large relative <strong>to</strong> (ap,r − ap,p), then there is more added benefit for a team<br />
<strong>to</strong> run aga<strong>in</strong>st a pass defense than it is <strong>to</strong> pass aga<strong>in</strong>st a run defense (relative<br />
<strong>to</strong> each play be<strong>in</strong>g carried out aga<strong>in</strong>st an appropriate defense). Interest<strong>in</strong>gly,<br />
and somewhat counter<strong>in</strong>tuitively, this bonus for a team <strong>to</strong> “guess right” with<br />
a runn<strong>in</strong>g play results <strong>in</strong> a smaller overall proportion of runn<strong>in</strong>g plays be<strong>in</strong>g<br />
called.
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