Unit-sphere games

This paper introduces a class of games, called unit-sphere games, in which strategies are real vectors with unit 2-norms (or, on a unit-sphere). As a result, they should no longer be interpreted as probability distributions over actions, but rather be thought of as allocations of one unit of resource to actions and the payoff effect on each action is proportional to the square root of the amount of resource allocated to that action. The new definition generates a number of interesting consequences. We first characterize the sufficient and necessary condition under which a two-player unit-sphere game has a Nash equilibrium. The characterization reduces solving a unit-sphere game to finding all eigenvalues and eigenvectors of the product matrix of individual payoff matrices. For any unit-sphere game with non-negative payoff matrices, there always exists a unique Nash equilibrium; furthermore, the unique equilibrium is efficiently reachable via Cournot adjustment. In addition, we show that any equilibrium in positive unit-sphere games corresponds to approximate equilibria in the corresponding normal-form games. Analogous but weaker results are obtained in n-player unit-sphere games.

1. According to certain existing empirical evaluations (see Agarwal et al. 2011), the CTR of an ad link is concave in payment, where the degree of concavity depends on the keyword length. In our example, the CTR is proportional to the square root of the money spent, which is one special and commonly used concave function. In fact, it is standard to assume that the cost of a certain amount of “effort” e (e.g., CTR in our example) is proportional to \(e^2\). See Holmstrom and Milgrom (1987), Hauser et al. (1994), Lafontaine and Slade (1996) and Hu et al. (2015).

2. In cases where \(Ay = 0\) (resp. \(Bx = 0\)), one may set \(\alpha \) (resp. \(\beta \)) arbitrarily.

3. We say a matrix \(A > 0\) if \(A_{ij} > 0\) for all (i, j), and a vector \(x \ge 0\) if \(x_i \ge 0\) for all i.

4. Recall that \(\rho (AB) = \rho (BA)\).

5. Linear convergence is another way of saying the error diminishes exponentially fast in the number of iterations.

6. A multiplicative k-approximate MSNE denotes a strategy profile where no player can improve her utility by k times via deviation.

We are grateful to Andrew Yao for helpful discussions. Part of this work is done while Pingzhong Tang was visiting Simons institute at UC Berkeley. This work was supported by the National Basic Research Program of China Grant 2011CBA00300, 2011CBA00301, the Natural Science Foundation of China Grant 61033001, 61361136003, 61303077, 61561146398, a Tsinghua Initiative Scientific Research Grant and a China Youth 1000-talent program.

Authors and Affiliations

1. IIIS, Tsinghua University, Beijing, China

   Pingzhong Tang & Hanrui Zhang

Corresponding author

Correspondence to Pingzhong Tang.

Reprints and permissions