Problem Set 1 Solutions

Transcription

1 Problem Set 1 Solutions Most of you did very well for your first problem set, good job! Extra kudos to teams responsible for the model solutions attached. Some comments: 1. Most decisions trees covered the binaries choices - offer/no offer and accept/reject - very well. However a decision tree should also show: the range of possible offers, e.g. between $1 and $10M for the incumbent at which node the offer amounts are decided See Sharif et al s solution for Q1 for an example. Remember also to parameterize offers by variables, e.g. incumbent offers X, where X is between $0 and $10M and rival offers Y, where Y is between $0 and $9.5M. Then express payoffs in terms of X and Y. See Wrecking Ball s Q1 for an example. Some of you modeled the rival as the dice and others as being able to choose any dollar amount. Both approaches are fine here. 2. To show that a set of actions and payoffs is a Nash Equilibrium you need to show that no agent has a profitable deviation. In this case you need to show that the equilibrium is optimal for the player, the incumbent, and the rival, given the actions of the other 2 parties. 3. Drawing decision trees by hand is fine, but please make sure they are clear. Some were very hard to read. Thanks, Sibo

2 MBA 211 Game Theory Team Wrecking Ball Guilherme Porto Florido Jana Beiswenger Jeff Routh Megan Bradfield Patrick Menendez Problem-solving questions: Questions 1. Assume that negotiations always take the form of a take-it-or-leave it offer by the firm followed by an acceptance or rejection from the agent. Note that in case of rejection, the offer remains on the table. Draw the game tree for the version of the game where there is no right of first refusal. 2. What is the equilibrium to this game?

3 The equilibrium of the game goes towards (0.5 ; 0 ; 9.5), as Team 1 will always offer 9.5 M to avoid having the Player accepting a potential higher offer from Team 2, which is highly likely to occur. In this situation, the Player has a game advantage. 3. Now show how the game tree changes with the addition of the right of first refusal. 4. How does the equilibrium change with the addition of a right of first refusal?

4 In this case the equilibrium changes to (9.99 ; 0 ; 0,01) or to a higher offer made initially by Team1 considering the high probability of Team 2 making no offer and the fact that Team 1 has the final call. The right of first refusal shifts the game advantage to Team 1 instead of Player. Discussion questions: 5. How valuable is the right of first refusal clause in a contract? Do you think this is appreciated by those entering into it? Is there some way the agent can change the system to undermine the power of this clause? Does it promote competition? The right of first refusal, as mentioned before, brings the advantage to Team 1. Due to this fact, players do not appreciate entering this clause. When entering into contracts, agents can try to change the system by negotiating a time period where the ROFR is valid. For example, ROFR is only valid for the first 5 years and after that, the player becomes an Unrestricted Free Agent, allowing him/her to sign with any team without the option of their existing team having the ROFR. This tips the balance towards the player s favor later on, but also incentivizes them to perform at their very best, promoting better interleague competition, for when the ROFR option is removed from their contract. Moreover, the right of first refusal is a resource for teams to keep their star players. This timeperiod clause ends up favoring stars versus regular players on negotiations with their current teams. To the league itself, the right of first refuse forces teams to comprise their budget to maintain star players in their rosters, which helps to balance the number of good players among the teams.

5 Sharif Karmally Andrew Meyers Qais Al Shihabi Moulay Driss Belkebir Mrani Juan Zarruk Game Theory Problem Set #1: Right of First Refusal 1) Payoffs written as ( Incumbent I, Player P, Rival R ) 2) This game can be solved using backward induction. In the final step, the Player will accept either the Rival s offer or the Incumbent s offer, whichever is greater. Since the Rival loses $0.5M if it makes an offer that is not accepted, the Rival prefers to not make an offer (and get a guaranteed payoff of $0M) if it cannot top the Incumbent s offer. In the second last step, the Rival is willing to top the incumbent s bid up to a maximum bid of $9.5M ($10M value of player less $0.5 costs of making an offer). In the second step, the Player always rejects the Incumbent s offer since it can choose to accept it later on (it stays on the table) and the Rival may be irrational and top the bid even if it loses money by doing so. In the first step, the Incumbent can now see that if it offers less than $9.5M, the Rival will top its offer and the Incumbent will finish with $0M. However, if the Incumbent offers $9.5M, the Rival will not make an offer and the Player will ultimately accept the Incumbent s offer: 1

6 Sharif Karmally Andrew Meyers Qais Al Shihabi Moulay Driss Belkebir Mrani Juan Zarruk 3) 4) Since the Incumbent can match the Rival s offer, it will do so whenever the Rival makes an offer, and the Player will always accept the Incumbent s revised offer. The Rival realizes this and is no longer willing to make an offer under any circumstance. With no competition, the Incumbent offers the player the lowest amount possible without offending the player so greatly that the player chooses to forego the salary to save his/her pride. This amount is set to $0.1M in the illustration below. 2

7 Sharif Karmally Andrew Meyers Qais Al Shihabi Moulay Driss Belkebir Mrani Juan Zarruk 5) The right of first refusal can be very valuable. Under the rules we set out in this game, the ROFR changed the game from the Player receiving $9.5M (his/her true value less the costs of a rival making an offer) to receiving the lowest possible value he would still accept, let s say $0.1M. All of this value is captured by the Incumbent, so the right of first refusal is worth around $9.4M. This is probably not always appreciated by those entering into it. They are likely convinced that the rationale for it is to give the team that has invested so much in them and whose fans are so attached to the player the opportunity to retain them for the same amount as they are being offered by a competitor. The effect on a competitor s willingness to make an offer is likely not considered by some. In other words, the clause reduces competition in the market for players, and this is not always appreciated. To undermine the power of this clause, a player could show very publicly that they are dissatisfied with their current team/manager/coach. This would indicate to Rivals that there is an intangible value that the Player would add to the Rival offer, and that the Player may accept the Rival offer even if it is matched by the Incumbent. Another way to undermine the power of this clause would be for the Player to offer to reimburse the expenses incurred in making an offer for any Rival who does so and is successful. This would level the playing field between the Rival and the Incumbent. Since the Incumbent gets to move last, it still may be able to match, so the Player would have to offer to reimburse some small amount over and above the expenses to truly undermine the clause. Finally, it is worth stating that a big assumption here is that all organizations, the Incumbent and any potential Rivals, all agree on the value of the player. If the incumbent has many star players, the incremental value of this player may be much less than for a Rival firm in desperate need of a star. In this case, the Rival firm would be willing to pay more and could offer an amount so high that the Incumbent would rather not exercise its right of first refusal. 3