In the paper “Comments on the Interpretations of Game Theory,” published in Volume 59, Issue 4 of Econometrica in July 1991, Ariel Rubinstein discusses notions of game theory and strategy that aims to highlight some of the inconsistency between the interpretation and application. Rubinstein argues that equilibrium strategy describes not only a player’s plan of action, but also the considerations that support the optimality of the plans. The paper also argues that models should encompass the perception of a real life situation by the decision makers. Game theory, as Rubinstein writes, is not simply about abstract mathematics but about the real world.
The first half of the paper deals with the notion of strategy. Rubinstein argues that the conventional interpretation is inconsistent with the way it is applied, leading to confusion. In one of the contexts, Rubinstein talks about extensive games with more than one moves. In the game, a player’s strategy is required to specify an action for each node in the game tree corresponding to that player’s movement. However, the incongruence comes when an action must be specified, even after earlier moves that would make the subsequent decision point inconsistent with the earlier moves. The necessity of this specification stems from the need to determine subgame-perfect equilibrium to test the rationality of the plans. In all, a strategy needs to encompass not only the player’s plan, but also the opponents’ belief in the chance that the plan was not followed.
Rubinstein also looks at the interpretation of strategy in mixed strategies. While intuitively problematic, there are clear cases in which players choose random actions, preferring over pure strategies. One interpretation of mixed strategy is to use a large population and having each occurrence take a random draw of the items from the population. Another interpretation is the purification idea, whereby a mixed strategy is dependent on private information not specified in the model. This interpretation argues that ostensibly random behaviors are actually deterministic. Finally, Rubinstein looks at the case of limited memory, which helps to keep modes of behavior simple. When probabilistic nature of doubt is introduced, there may be additional decision points that are unreachable, but added to the game form merely to allow the discussion of reasoning in the state of doubt.
From these discussions, Rubinstein tries to adopt the view that a game is not “a rigid description of the physical rules of the world,” but instead a “comprehensive description of the relevant factors involved in a specific situation as perceived by the players.” Towards that idea, model should include only the factors perceived by the players to be relevant, making the application of game theory “more of an art than a mechanical algorithm.” Finally, Rubinstein talks about regularity, which is necessary for employing game theory as a descriptive science. Game theory, in conclusion, builds models from intuition and from mathematical knowledge uses deductive arguments, which can’t discover the truth alone. Instead, game theory also deals with psychological elements, which help to distinguish humans from machines, which Rubinstein believes is “more exciting and certainly more meaningful.”
Showing posts with label Game Theory. Show all posts
Showing posts with label Game Theory. Show all posts
Sunday, November 24, 2013
Tuesday, August 21, 2012
Showcase Showdown Analysis: Circular Reasoning and Oscillating Nash Equilibrium
The Showcase Showdown is a portion of the game-show The Price Is Right, in which the contestants spin the Big Wheel that has 20 sections randomly distributed, from 5 cents to $1.00 in 5-cent increments. The objective is to get as close to $1.00 as possible without going over, with one initial spin and an optional second spin. In the game, three contestants play the game to determine who has the highest value. What's the strategy in this game for the first contestant? An intuitive response may be to spin again for 50 or less on the first spin; keep the first spin if it's 55 or greater. Unfortunately, it's not as simple as that.
One strategy would be to spin the wheel the second time, only if the first spin resulted in less than the expected total outcome from the game, conditional to the outcome from the first spin. We can analyze the extreme discrete cases first to get a better understanding. If the first spin is 100, all second spins will make the total go over, resulting in total score of 0. In that case, the contestant will definitely keep the first spin of 100. On the other extreme, the expected total from the two spins, conditional to the first being 5, is 52.25. In that case, taking the second spin is better off. Here is the complete table, with the higher result reflecting the course of action pursued:
If the first spin were 40 or less, having a second spin will on average produce a better result. The average of the third column gives the expected result from the game: 63. However, there is one hole in this reasoning when applied to the game. This would work perfectly fine if the first contestant played the game for himself or herself, only concerned about maximizing the individual score given the risk-reward offset. Instead, of the three contestants, only the one with the highest result wins. If the first contestant got anything from 45 to 60, inclusive on the first spin, it would've been strategically better to keep it in the aforementioned reasoning. However, since the objective is to beat all other contestants, rather than maximizing individual scores, at that point of the game it may be more reasonable to spin again nevertheless.
This is where the circular reasoning kicks in. If the contestant's decision is to spin the wheel the second time if the first spin were less than 63, then the total expected value drops to 59.95. Essentially, while the contestant tries to base the individual decision given the overall expected result, the overall expected results depend exactly on the individual contestants' decision. It's a circular route of logic, and also illustrates game theory being applied. Having 40 and 45 be the cutoff would the best risk-reward optimization decision on the individual level. However when that is the baseline, the dominant strategy is then to use 63 as the decision's critical point. When players do that, they all incur more risk and distort the overall expected result downward. Would contestant then use 59.95 as the decision's critical point?
If the contestants did, the next critical point would be 61.3. Here, if 61.3 were the next critical point, it forces 59.95 to again be the overall expected value, just like 63 did. Therefore, the "Nash equilibrium" is a perpetual oscillation between 59.95 and 61.3. In the end, the only definitely conclusions are to spin again if the first spin is 40 or lower, and to keep the first spin if it's 65 or greater. Having 45 and 50 as the first spin stands in the grey area, and 60 is dead in no man's land, caught in between the oscillating equilibrium.
One strategy would be to spin the wheel the second time, only if the first spin resulted in less than the expected total outcome from the game, conditional to the outcome from the first spin. We can analyze the extreme discrete cases first to get a better understanding. If the first spin is 100, all second spins will make the total go over, resulting in total score of 0. In that case, the contestant will definitely keep the first spin of 100. On the other extreme, the expected total from the two spins, conditional to the first being 5, is 52.25. In that case, taking the second spin is better off. Here is the complete table, with the higher result reflecting the course of action pursued:
| First Spin | 2 Spins Avg Total | Higher Result |
| 5 | 52.25 | 52.25 |
| 10 | 51.75 | 51.75 |
| 15 | 51 | 51 |
| 20 | 50 | 50 |
| 25 | 48.75 | 48.75 |
| 30 | 47.25 | 47.25 |
| 35 | 45.5 | 45.5 |
| 40 | 43.5 | 43.5 |
| 45 | 41.25 | 45 |
| 50 | 38.75 | 50 |
| 55 | 36 | 55 |
| 60 | 33 | 60 |
| 65 | 29.75 | 65 |
| 70 | 26.25 | 70 |
| 75 | 22.5 | 75 |
| 80 | 18.5 | 80 |
| 85 | 14.25 | 85 |
| 90 | 9.75 | 90 |
| 95 | 5 | 95 |
| 100 | 0 | 100 |
If the first spin were 40 or less, having a second spin will on average produce a better result. The average of the third column gives the expected result from the game: 63. However, there is one hole in this reasoning when applied to the game. This would work perfectly fine if the first contestant played the game for himself or herself, only concerned about maximizing the individual score given the risk-reward offset. Instead, of the three contestants, only the one with the highest result wins. If the first contestant got anything from 45 to 60, inclusive on the first spin, it would've been strategically better to keep it in the aforementioned reasoning. However, since the objective is to beat all other contestants, rather than maximizing individual scores, at that point of the game it may be more reasonable to spin again nevertheless.
This is where the circular reasoning kicks in. If the contestant's decision is to spin the wheel the second time if the first spin were less than 63, then the total expected value drops to 59.95. Essentially, while the contestant tries to base the individual decision given the overall expected result, the overall expected results depend exactly on the individual contestants' decision. It's a circular route of logic, and also illustrates game theory being applied. Having 40 and 45 be the cutoff would the best risk-reward optimization decision on the individual level. However when that is the baseline, the dominant strategy is then to use 63 as the decision's critical point. When players do that, they all incur more risk and distort the overall expected result downward. Would contestant then use 59.95 as the decision's critical point?
If the contestants did, the next critical point would be 61.3. Here, if 61.3 were the next critical point, it forces 59.95 to again be the overall expected value, just like 63 did. Therefore, the "Nash equilibrium" is a perpetual oscillation between 59.95 and 61.3. In the end, the only definitely conclusions are to spin again if the first spin is 40 or lower, and to keep the first spin if it's 65 or greater. Having 45 and 50 as the first spin stands in the grey area, and 60 is dead in no man's land, caught in between the oscillating equilibrium.
Wednesday, October 5, 2011
Livery Cab Bill Debate
Yellow taxis have become an icon of New York City, but their concentration around central Manhattan and the airports is undeniable. Many have complained that it's nearly impossible to catch a cab elsewhere in the city, and it's not difficult to note the paucity of the yellow cabs in the so-called outer boroughs. Currently, only yellow cabs are authorized to pick up passengers on the street who hail for service. To address this issue, a bill has been sitting at Governor Andrew Cuomo's desk that would authorize livery (black) cabs to pick up street hails in the outer boroughs. Currently, livery cabs pick up passengers on pre-arranged trips.
Aside from the access issue, proponents argue that this would generate millions for the city. Currently, drivers of the traditional yellow cabs spend hundreds of thousands of dollars to buy medallions, their exclusive right to pick up street hails within the city. By extending these medallion permits to livery cabs, "the tax revenues ... would be something like a billion dollars into the city’s budget," Mayor Bloomberg remarks.
Opposition comes from current yellow cab drivers, who argue that by leasing the right to pick up street hails to livery cabs, the value of their own medallions will be vastly diminished. On Tuesday, hundreds of cab drivers protested outside Governor Cuomo's office, urging him to veto the bill that would allow the livery cabs' operations. Furthermore, oppositions argue that "livery drivers in the outer boroughs would ignore prearranged pickups in favor of street hails."
Governor Cuomo has remarked that “the optimum goal is to design a plan that provides taxi access to the outer boroughs, access to the disabled, revenue for the city, and respects the medallion franchise.” The root cause of the problem is the current lack of cabs in outer boroughs. This disproportion is vividly illustrated by the fact that while 80% of the city's population lives outside Manhattan, 97% of the pickups are in central Manhattan or at the city's two airports, according to GPS data collected by taxi commission.
Most yellow cabs drivers stay within the airports and central Manhattan, where the demand for the cabs is the greatest. The situation is a microcosm of game theory practice. Each driver's dominant strategy is to stay in these hot-spots, for they're more likely to earn more revenue. However, if most drivers practice this method, all of them may be worse off, given that government intervention may deal with the lack of cabs in outer boroughs, through measures like authorizing the livery cabs, which would indirectly devalue all cab drivers' medallions. To settle this uneasy dilemma, an equilibrium needs to be sought that balances the desire of cab drivers to serve hot spots for more revenue, and the access for outer boroughs for citizens there. Within outer boroughs, secondary hot spots can be identified, including trains stations, shopping malls, sports venues, and other popular places. Monetarily incentives could be given to yellow cab drivers to pick up passengers from these "secondary hot spots" in the outer boroughs to compensate for their service in these less-profitable regions. This internal solution, if operational, may eradicate the need for external livery cabs and their rights to pick up street hails in the outer boroughs.
Sources:
Aside from the access issue, proponents argue that this would generate millions for the city. Currently, drivers of the traditional yellow cabs spend hundreds of thousands of dollars to buy medallions, their exclusive right to pick up street hails within the city. By extending these medallion permits to livery cabs, "the tax revenues ... would be something like a billion dollars into the city’s budget," Mayor Bloomberg remarks.
Opposition comes from current yellow cab drivers, who argue that by leasing the right to pick up street hails to livery cabs, the value of their own medallions will be vastly diminished. On Tuesday, hundreds of cab drivers protested outside Governor Cuomo's office, urging him to veto the bill that would allow the livery cabs' operations. Furthermore, oppositions argue that "livery drivers in the outer boroughs would ignore prearranged pickups in favor of street hails."
Governor Cuomo has remarked that “the optimum goal is to design a plan that provides taxi access to the outer boroughs, access to the disabled, revenue for the city, and respects the medallion franchise.” The root cause of the problem is the current lack of cabs in outer boroughs. This disproportion is vividly illustrated by the fact that while 80% of the city's population lives outside Manhattan, 97% of the pickups are in central Manhattan or at the city's two airports, according to GPS data collected by taxi commission.
Most yellow cabs drivers stay within the airports and central Manhattan, where the demand for the cabs is the greatest. The situation is a microcosm of game theory practice. Each driver's dominant strategy is to stay in these hot-spots, for they're more likely to earn more revenue. However, if most drivers practice this method, all of them may be worse off, given that government intervention may deal with the lack of cabs in outer boroughs, through measures like authorizing the livery cabs, which would indirectly devalue all cab drivers' medallions. To settle this uneasy dilemma, an equilibrium needs to be sought that balances the desire of cab drivers to serve hot spots for more revenue, and the access for outer boroughs for citizens there. Within outer boroughs, secondary hot spots can be identified, including trains stations, shopping malls, sports venues, and other popular places. Monetarily incentives could be given to yellow cab drivers to pick up passengers from these "secondary hot spots" in the outer boroughs to compensate for their service in these less-profitable regions. This internal solution, if operational, may eradicate the need for external livery cabs and their rights to pick up street hails in the outer boroughs.
Sources:
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