Can a player change the game?
Consider the kidnapping situation in the movie Ransom. Mel Gibson plays the character Tom Mullen, whose son, Sean, is kidnapped by a corrupt police officer named Jimmy Shaker. Initially, the situation operates like a standard kidnapping. Shaker demands a 2 million dollar ransom. While going to make the ransom drop, Tom Mullen becomes convinced that the kidnappers have no intention of releasing his son. He then decides to go to the local television sta- tion and is filmed live making the following announcement:
The whole world now knows my son, Sean Mullen, was kidnapped, for ran- som, three days ago. This is a recent photograph of him. Sean, if you’re watching, we love you. And this is what waits for the man that took him. This is your ransom. Two million dollars in unmarked bills, just like you wanted. But this is as close as you’ll ever get to it. You’ll never see one dol- lar of this money, because no ransom will ever be paid for my son. Not one dime, not one penny. Instead, I’m offering this money as a reward on your head. Dead or alive, it doesn’t matter. So congratulations, you’ve just become a 2 million dollar lottery ticket, except the odds are much, much better. Do you know anyone that wouldn’t turn you in for 2 million dollars? I don’t think you do. But, this is your last chance, if you return my son, alive, un- injured, I’ll withdraw the bounty. With any luck, you can simply disappear. Understand. You will never see this money. Not one dollar. So you still have a chance to do the right thing. If you don’t, well, then, God be with you, be- cause nobody else on this Earth will be.
It is clear that Tom Mullen had a “brainstorm” of converting the ransom into a bounty. Furthermore, it is natural to suppose that this possibility was
2.9 A Few More Issues in Modeling Games 47
not one that Jimmy Shaker had considered. In a way, the players started with the game looking like that in Figure 2.1, but then Tom Mullen “changed” the game to something else. Unfortunately, game theory does not allow for such changes or innovations. A key assumption of game theory is that the game is initially understood and agreed upon by the players; the rules of the game are common knowledge.
What we can do, however, is modify the game to that in FIGURE 2.16. John Mullen now has the option of paying the ransom, offering a bounty, or doing nothing, in response to each of which Jimmy Shaker has the two options of releasing or killing Sean Mullen. This game is understood by the players when they start it and thus does not allow for the possibility of Tom Mullen “sur- prising” Jimmy Shaker by offering a bounty. True innovation is not a feature that current game theory can encompass. Thus, the answer is that players are not allowed to change the game. However, we can always enrich the game and give players more options.
FIGURE 2.16 Extensive Form for the Film Ransom
John Mullen
7
Kill
3
Jimmy Shaker
Jimmy Shaker
JS JS JS
John Mullen
6
4
Release
Do not pay ransom
Pay ransom
Offer bounty
5
Do not kidnap
Kidnap
7
4
Kill
1
3
6
Release
1
Kill
2
2
5
Release
Consider the kidnapping situation in the movie Ransom. Mel Gibson plays the character Tom Mullen, whose son, Sean, is kidnapped by a corrupt police officer named Jimmy Shaker. Initially, the situation operates like a standard kidnapping. Shaker demands a 2 million dollar ransom. While going to make the ransom drop, Tom Mullen becomes convinced that the kidnappers have no intention of releasing his son. He then decides to go to the local television sta- tion and is filmed live making the following announcement:
The whole world now knows my son, Sean Mullen, was kidnapped, for ran- som, three days ago. This is a recent photograph of him. Sean, if you’re watching, we love you. And this is what waits for the man that took him. This is your ransom. Two million dollars in unmarked bills, just like you wanted. But this is as close as you’ll ever get to it. You’ll never see one dol- lar of this money, because no ransom will ever be paid for my son. Not one dime, not one penny. Instead, I’m offering this money as a reward on your head. Dead or alive, it doesn’t matter. So congratulations, you’ve just become a 2 million dollar lottery ticket, except the odds are much, much better. Do you know anyone that wouldn’t turn you in for 2 million dollars? I don’t think you do. But, this is your last chance, if you return my son, alive, un- injured, I’ll withdraw the bounty. With any luck, you can simply disappear. Understand. You will never see this money. Not one dollar. So you still have a chance to do the right thing. If you don’t, well, then, God be with you, be- cause nobody else on this Earth will be.
It is clear that Tom Mullen had a “brainstorm” of converting the ransom into a bounty. Furthermore, it is natural to suppose that this possibility was
2.9 A Few More Issues in Modeling Games 47
not one that Jimmy Shaker had considered. In a way, the players started with the game looking like that in Figure 2.1, but then Tom Mullen “changed” the game to something else. Unfortunately, game theory does not allow for such changes or innovations. A key assumption of game theory is that the game is initially understood and agreed upon by the players; the rules of the game are common knowledge.
What we can do, however, is modify the game to that in FIGURE 2.16. John Mullen now has the option of paying the ransom, offering a bounty, or doing nothing, in response to each of which Jimmy Shaker has the two options of releasing or killing Sean Mullen. This game is understood by the players when they start it and thus does not allow for the possibility of Tom Mullen “sur- prising” Jimmy Shaker by offering a bounty. True innovation is not a feature that current game theory can encompass. Thus, the answer is that players are not allowed to change the game. However, we can always enrich the game and give players more options.
FIGURE 2.16 Extensive Form for the Film Ransom
John Mullen
7
Kill
3
Jimmy Shaker
Jimmy Shaker
JS JS JS
John Mullen
6
4
Release
Do not pay ransom
Pay ransom
Offer bounty
5
Do not kidnap
Kidnap
7
4
Kill
1
3
6
Release
1
Kill
2
2
5
Release