Direct Match Cooperation
Are people cooperative airplane passengers? If so, to what extent and with whom do we cooperate?
It depends who you ask. Your own answer might depend on who and what they ask of you as well as what you get in return. Of course, we cooperate with airline employees, airport security staff, law enforcement, and so on. That seems intuitive in public interactions. Think instead about cooperation in terms of which passenger you ask when you want to move to another seat.
Our reseating word problem uses 6 statements from which we determine exchange probability. You be "Passenger X" and let another person be "Passenger Y". Each passenger describes their preference in the following statements.
I need a _______ seat near the ______ of the aircraft.
My priority is to move in the __________ direction.
I need a _______ seat near the ______ of the aircraft.
My priority is to move in the __________ direction.
The priority statement indicates willingness to accept a tradeoff in move direction. Your move priority is relative to your currently assigned seat and where the other seat should be: forward or backward (i.e. North or South), or left or right (i.e. West or East), of your seat.
Hopscotch maps those statements into a unified math model of reciprocal valuation. We rank suitability of a match to both passengers. This represents the simultaneous exchange feasibility in the two-sided valuation model. Next, we calculate the score for potential cooperation.
"Figure 1: Seat Match Quadrants" below shows a two passenger exchange plot . The left graph shows the hypothetical goodness of fit between two passengers considering a seat exchange. The right graph illustrates the exchange quadrants 1 through 4.
The left graph is labeled "Fitness of Seat" for these two passengers. The right graph is labeled "Reciprocal Seat Match Quadrants". Notice we can ignore actual locations of both seats. This was not a mistake or magic, it was algebraic transformation. Let's look at our Quadrant 2 exchange model.
Starting from the origin, at the crosshairs in the center, move horizontally to the right along the Passenger X-axis. Notice the goodness of fit line increases linearly and gradually up to the point labeled with 2 orange smiley-faces. Similarly, start at the origin and trace upwards along the Passenger Y-axis and notice the reciprocal fitness (equal to X) for Passenger Y.
We created two bounded boxes formed by the lines on the graph. Four lines form these boxed areas and are labeled with smiley-faces on the axes. The lines intersect to form the box corners. The box corners also intersect on the tip of 2 teardrop-shaped green regions.
These green regions were "flattened" or squashed from their z-axis coming out of the page. The height of the green region, if shown, would represent the fit quality. The two-dimensional plots show more smiley faces to mean the green region pushes farther out of the page. Hopscotch calculates the height of the green area as "fit quality".
Lines forming box 1 has:
Passenger X= 😀😀
Passenger Y= 😀😀
Lines forming box 2 has:
Passenger X= 😀😀😀
Passenger Y= 😀😀😀
It depends who you ask. Your own answer might depend on who and what they ask of you as well as what you get in return. Of course, we cooperate with airline employees, airport security staff, law enforcement, and so on. That seems intuitive in public interactions. Think instead about cooperation in terms of which passenger you ask when you want to move to another seat.
Reseating Mad Lib
Do you remember the "Mad Lib" party game of completing a sentences to make a (usually funny) story? Suppose there exists two passengers, each needing to move to another seat. They know nothing about one another (i.e. current seat or preference). Their reason and purpose could be very different from one another; exact opposites even.Our reseating word problem uses 6 statements from which we determine exchange probability. You be "Passenger X" and let another person be "Passenger Y". Each passenger describes their preference in the following statements.
Passenger X
I have a _______ seat near the ______ of the aircraft.I need a _______ seat near the ______ of the aircraft.
My priority is to move in the __________ direction.
Passenger Y
I have a _______ seat near the ______ of the aircraft.I need a _______ seat near the ______ of the aircraft.
My priority is to move in the __________ direction.
The priority statement indicates willingness to accept a tradeoff in move direction. Your move priority is relative to your currently assigned seat and where the other seat should be: forward or backward (i.e. North or South), or left or right (i.e. West or East), of your seat.
Hopscotch maps those statements into a unified math model of reciprocal valuation. We rank suitability of a match to both passengers. This represents the simultaneous exchange feasibility in the two-sided valuation model. Next, we calculate the score for potential cooperation.
"Figure 1: Seat Match Quadrants" below shows a two passenger exchange plot . The left graph shows the hypothetical goodness of fit between two passengers considering a seat exchange. The right graph illustrates the exchange quadrants 1 through 4.
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| Figure 1: Seat Match Quadrants for 2 Passengers |
The left graph is labeled "Fitness of Seat" for these two passengers. The right graph is labeled "Reciprocal Seat Match Quadrants". Notice we can ignore actual locations of both seats. This was not a mistake or magic, it was algebraic transformation. Let's look at our Quadrant 2 exchange model.
Quadrant 2: Exact Match Feasibility
Consider the top right panel in the "Fitness of Seat" plot (the Fig. 1 graph on the left). These passenger's mirror their affinity for the other person's seat along a 45-degree axis. This reciprocal equivalence is a special case where both passengers view the other passenger's seat in an equally positive light (i.e. the seat assigned to the other person). We added visual cues on the x and y-axis as color-coded smiley-faces representing satisfaction levels of Passenger X, in orange, and Passenger Y in blue.Starting from the origin, at the crosshairs in the center, move horizontally to the right along the Passenger X-axis. Notice the goodness of fit line increases linearly and gradually up to the point labeled with 2 orange smiley-faces. Similarly, start at the origin and trace upwards along the Passenger Y-axis and notice the reciprocal fitness (equal to X) for Passenger Y.
We created two bounded boxes formed by the lines on the graph. Four lines form these boxed areas and are labeled with smiley-faces on the axes. The lines intersect to form the box corners. The box corners also intersect on the tip of 2 teardrop-shaped green regions.
These green regions were "flattened" or squashed from their z-axis coming out of the page. The height of the green region, if shown, would represent the fit quality. The two-dimensional plots show more smiley faces to mean the green region pushes farther out of the page. Hopscotch calculates the height of the green area as "fit quality".
Lines forming box 1 has:
Passenger X= 😀😀
Passenger Y= 😀😀
Lines forming box 2 has:
Passenger X= 😀😀😀
Passenger Y= 😀😀😀
Why are there multiple fit quality regions (the intersecting areas on the green teardrop shapes)? Because "Figure 1" illustrates three distinct seat scenarios on a single graph. Your move priority can be to move north and south or east and west. "Figure 2: Seat Move Direction Priority" below shows move priority direction.
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| Figure 2: Seat Move Direction Priority |
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