Strategic behavior in non-atomic games

In order to remedy the possible loss of strategic interaction in non-atomic games with a societal choice, this study proposes a refinement of Nash equilibrium, strategic equilibrium. Given a non-atomic game, its perturbed game is one in which every player believes that he alone has a small, but positive, impact on the societal choice; and a distribution is a strategic equilibrium if it is a limit point of a sequence of Nash equilibrium distributions of games in which each player's belief about his impact on the societal choice goes to zero. After proving the existence of strategic equilibria, we show that all of them must be Nash. Moreover, it is displayed that in many economic applications, the set of strategic equilibria coincides with that of Nash equilibria of large finite games

Strategic Behavior in Non-Atomic Games

Typically, economic situations featuring a large number of agents are not modelled with a finite normal form game, rather by a non-atomic game. Consequently, the possibility of strategic interaction may be completely ignored. In order to restore strategic interaction among agents we propose a refinement of Nash equilibrium, strategic equilibrium, for non-atomic games with a continuum of agents, each of whose payo® depends on what he chooses and a societal choice. Given a non-atomic game, we consider a perturbed game in which every player believes that he alone has a small, but positive, impact on the societal choice. A strategy profile is a strategic equilibrium if it is a limit point of a sequence of Nash equilibria of games in which each player's belief about his impact on the societal choice goes to zero. After proving the existence of strategic equilibria, we show that every strategic equilibrium must be a Nash equilibrium of the original non-atomic game, thus, our concept of strategic equilibrium is indeed a refinement of Nash equilibrium. Next, we show that the concept of strategic equilibrium is the natural extension of Nash equilibrium infinite normal form games, to non-atomic games: That is, given any finite normal form game, we consider its non- atomic version, and prove that a strategy profile, in the non-atomic version of the given finite normal form game, is a strategic equilibrium if and only if the associated strategy profile in the finite form game is a Nash equilibrium. Finally, applications of strategic equilibrium is presented examples in which the set of strategic equilibria, in contrast with the set of Nash equilibria, does not contain any implausible Nash equilibrium strategy profiles. These examples are: a game of proportional voting, a game of allocation of public resources, and finally non-atomic Cournot oligopoly

One - Memory in Repeated Games

We study the extent to which equilibrium payo®s of discounted repeated games can be obtained by 1 { memory strategies. First, we present robust examples of games in which there is a subgame perfect equilibrium payo® pro¯le that cannot be obtained by any 1 { memory subgame perfect equilibrium. Then, a complete characterization of 1 { memory simple strategies is provided, and it is employed to establish the following in games with more than two players each having connected action spaces: 1. all subgame perfect equilibrium payo®s can be approximately supported by an " { sub- game perfect equilibrium strategy of 1 { memory, 2. all strictly enforceable subgame perfect equilibrium payo®s can be approximately sup- ported by a 1 { memory subgame equilibrium, and 3. the subgame perfect Folk Theorem holds for 1 { memory strategies. While no further restrictions are needed for the third result to hold in 2 { player games, an additional restriction is needed for the ¯rst two: players must have common punishments.

Optimality of linearity with collusion and renegotiation

This study analyzes a continuous-time N-agent Brownian hidden-action model with exponential utilities, in which agents' actions jointly determine the mean and the variance of the outcome process. In order to give a theoretical justi¯cation for the use of linear contracts, as in Holmstrom and Milgrom (1987), we consider a variant of its generalization given by Sung (1995), into which collusion and renegotiation possibilities among agents are incorporated. In this model, we prove that there exists a linear and stationary optimal compensation scheme which is also immune to collusion and renegotiation.Principal-agent problems; moral hazard; linear contracts; continuous-time model; Brownian motion martingale method; collusion,; renegotiation; team

Strategic behavior in non-atomic games

In order to remedy the possible loss of strategic interaction in non-atomic games with a societal choice, this study proposes a refinement of Nash equilibrium, strategic equilibrium. Given a non-atomic game, its perturbed game is one in which every player believes that he alone has a small, but positive, impact on the societal choice; and a distribution is a strategic equilibrium if it is a limit point of a sequence of Nash equilibrium distributions of games in which each player's belief about his impact on the societal choice goes to zero. After proving the existence of strategic equilibria, we show that all of them must be Nash. Moreover, it is displayed that in many economic applications, the set of strategic equilibria coincides with that of Nash equilibria of large finite games.Strategic equilibrium; Games with a continuum of players; Equilibrium distributions

Time Dependent Bounded Recall Strategies Are Enough to Play the Discounted Repeated Prisoners Dilemma

We show that for any discount factor, there is a natural number M such that all subgame perfect equilibrium outcomes of the discounted repeated prisoners dilemma can be obtained by subgame perfect equilibrium strategies with the following property: current play depends only on the number of the time-index and on the history of the last M periods. Therefore, players who are restricted to using pure strategies, have to remember, at the most, M periods in order to play any equilibrium outcome of the discounted repeated prisoners dilemma. This result leads us to introduce the notion of time dependent complexity, and to conclude that in the repeated prisoners dilemma, restricting attention to finite time dependent complex strategies is enough.

Banks versus venture capital when the venture capitalist values private benefits of control

If control of their firms allows entrepreneurs to derive private benefits, it also allows other controlling parties. Private benefits are especially relevant for venture capitalists, who typically get considerable control in their portfolio firms, but not for banks, which are passive loan providers. We incorporate this difference between banks and venture capital and analyze entrepreneurs' financing strategy between the two. We find that, in all strict Nash Equilibria, entrepreneurs who value private benefits more choose banks while the rest choose venture capital. Thus, bank-financed entrepreneurs allocate more resources to tasks that yield private benefits while VC-backed entrepreneurs have higher profitability.bank, control, entrepreneurship, private benefit, venture capital

Time Dependent Bounded Recall Strategies Are Enough to Play the Discounted Repeated Prisoners' Dilemma

We show that for any discount factor, there is a natural number $M$ such that all subgame perfect equilibrium outcomes of the discounted repeated prisoners' dilemma can be obtained by subgame perfect equilibrium strategies with the following property: current play depends only on the number of the time-index and on the history of the last $M$ periods. Therefore, players who are restricted to using pure strategies, have to remember, at the most, $M$ periods in order to play any equilibrium outcome of the discounted repeated prisoners' dilemma. This result leads us to introduce the notion of time dependent complexity, and to conclude that in the repeated prisoners' dilemma, restricting attention to finite time dependent complex strategies is enough.Repeated Prisoners' Dilemma; Memory; Bounded Rationality

Investor's increased shareholding due to entrepreneur-manager collusion

This study presents an investor/entrepreneur model in which the entrepreneur has opportunities to manipulate the workings of the project via hidden arrangements. We provide the optimal contracts in the presence and absence of such hidden arrangements. The contracts specify the shareholding arrangement between investor and entrepreneur. Moreover, we render an exact condition necessary for the credit market to form

Optimality of linearity with collusion and renegotiation

This study analyzes a continuous-time N-agent Brownian hidden-action model with exponential utilities, in which agents' actions jointly determine the mean and the variance of the outcome process. In order to give a theoretical justification for the use of linear contracts, as in Holmstrom and Milgrom (1987), we consider a variant of its generalization given by Sung (1995), into which collusion and renegotiation possibilities among agents are incorporated. In this model, we prove that there exists a linear and stationary optimal compensation scheme which is also immune to collusion and renegotiation