# The value of a draw

## Abstract

We model a match as a recursive zero-sum game with three possible outcomes: Player 1 wins, player 2 wins, or there is a draw. We focus on matches whose point games also have three possible outcomes: Player 1 scores the point, player 2 scores the point, or the point is drawn in which case the point game is repeated. We show that a value of a draw can be attached to each state so that an easily computed stationary equilibrium exists in which players’ strategies can be described as minimax behavior in the point games induced by these values.

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1. 1.

The game of chess itself is also a match. In fact, in the first article to appear on game theory, Zermelo (1913) models chess as a zero-sum recursive game.

2. 2.

The 1984 Karpov–Kasparov match lasted five months and was aborted after 48 games when the partial score was 5–3. Coincidentally, the longest penalty shootout to date also had 48 kicks.

3. 3.

In his article on the game of chess, Zermelo (1913) states: “Such a possible endgame q can find its natural end in a ‘checkmate’ ..., but could also – at least theoretically – go on forever, in which case the game would without doubt have to be called a draw ...” (see Schwalbe and Walker 2001).

4. 4.

We are aware that in real chess, the outcome of a pair of strategies is deterministic. We hope chess enthusiasts will forgive this distortion.

5. 5.

If a state k had only one proper successor, we could treat k as the missing proper successor and denote these successors by w(k) and $$\ell (k)$$. The matrix $$P^k(\varepsilon )$$ would then be defined as $$\{p_{ij}^{kw(k)}|i\in {{\mathcal {I}}},\, j\in {\mathcal {J}} \}$$, and with this amended definition, the ensuing analysis would remain valid.

6. 6.

We should note, though, that for binary matches the above-mentioned invariance does hold. That is, if $${\varGamma }'_\lambda$$ is a binary discounted match that is obtained from the binary discounted match $${\varGamma }_{\lambda }$$ by slightly perturbing the nonzero entries of its point matrices such that for state k , $$v^{w(k)}_\lambda> v^{\ell (k)}_\lambda \Leftrightarrow {v'}^{w(k)}_{\lambda } > {v'}^{\ell (k)}_{\lambda }$$, then equilibrium behavior in state k is the same in both $${\varGamma }_\lambda$$ and $${\varGamma }'_{\lambda }$$.

7. 7.

There is an alternative way of building an equilibrium labeling, according to which w(s) is chosen to be a successor of s with the highest discounted value for a sufficiently high discount factor. Figuring out, however, the appropriate discount factor and the associated discounted value is not an easy task.

8. 8.

There may be more than one natural labeling. For our analysis, any of them will do.

9. 9.

A set C is a recurrent class if $$\sum _{k'\in C} \mu ^{kk'}(\mathbf {x}^*, \mathbf {y})=1$$ for all $$k\in C$$ and no proper subset of C has this property. A state is transient if there is a positive probability of leaving and never returning.

10. 10.

This game resembles a chess position in which players can either transition to an endgame with unfavorable odds or keep playing safe. It is usually the case that players prefer the second option, and thus, the game ends in a draw.

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## Acknowledgements

Oscar Volij thanks the Department of Foundations of Economics Analysis I at the University of the Basque Country for its kind hospitality. This paper was partly written during his stay there.

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Correspondence to Oscar Volij.