Game Theory is a tool used in several fields of science including Political Science and Economics. For this article a very simplified version will be used where the model consists of a rational agent (it always chooses to perform the action with the optimal expected outcome for itself from among all feasible actions) competing in a mathematically defined competition.
To put it more plainly, we are going to try and find the theoretical scoring maximums for each part of the game.
Using a game theory based evaluation of the scoring system accomplishes three things:
- Makes sure the students understand the scoring rules
- Positions optimized scoring strategies at the front of the discussion leading into system strategy brainstorming
- Provides the team with an objective reference that can be used as an unbiased evaluation of strategies in later phases of design
To begin, we will break the game into three sections for evaluation:
- Auto (15 Seconds)
- Teleoperation Mode (135 Seconds)
- End Game (T-minus 30 Seconds).
In each section we ask: What are all of the ways our robot can score points? Are any of these scoring methods mutually exclusive (for example, we can only score a single POWER CELL in one of the three POWER PORT levels)? Do any of the scoring methods share casual relationships (for example, nine (9) POWER CELLS must be scored before a CONTROL PANEL can be rotated)?
For reference we will add a copy of “Figure 3-3 INFINITE RECHARGE FIELD” from the game manual to clarify location terminology.
In Autonomous Mode, teams stage their ROBOT on their INITIATION LINE.
The game manual (Table 4-2 Point Values) details that a robot can score points in this mode by:
- INITIATION LINE (5 points)
- POWER CELL scored in BOTTOM PORT (2 points)
- POWER CELL scored in OUTER PORT (4 points)
- POWER CELL scored in INNER PORT (6 points)
A robot can be pre-loaded with up to three (3) POWER CELLS (per Section 7.1 Rule G1.E in the Game Manual).
The first decision we need to make is what should we do with our POWER CELLS.
Treating the robot as a rational agent, it chooses to perform the actions with the optimal expected outcomes and scores three (3) POWER CELLS in the INNER PORT (18 pts), then moves off of the INITIATION LINE (5 pts) towards the SHIELD GENERATOR.
First, we assume it takes 3s to score the (3) POWER CELLS we start the match with, scoring 18pts. Next, we assume it takes 1s to move from the INITIATION LINE to the SHIELD GENERATOR, which scores 5pts. If we assume an average speed of 10 ft/s for the robot, then evaluate the distance to the SHIELD GENERATOR (~7ft), it takes approximately 1s to reach the SHIELD GENERATOR. Next, we assume it takes 5s to collect the POWER CELLS placed on the boundary of the RENDEZVOUS POINT. That puts us at 9s of AUTO used with 23 pts scored.
Choosing the shortest path to scoring, we turn and drive to our previous location on the INITIATION LINE. We travel to the INITIATION LINE (~7ft) which will take 1s, then shoot our (5) POWER CELLS (which we will assume takes 5s) scoring us 30pts. This takes 6s which finishes the total of our 15s in AUTO.
This MAX* Autonomous Mode achieves a score of 53 pts (count of 8 cells).
In Teleoperation Mode, the robot is located at the INITIATION LINE from the end of AUTO. The game manual (Table 4-2 Point Values) details that a robot can score points in this mode by:
- POWER CELL scored in BOTTOM PORT (1 points)
- POWER CELL scored in OUTER PORT (2 points)
- POWER CELL scored in INNER PORT (3 points)
- CONTROL PANEL with ROTATIONAL (10 points)
- Note: ROTATIONAL requires STAGE ONE (9 POWER CELLS)
- CONTROL PANEL with POSITIONAL* (20 points)
- Note: POSITIONAL requires STAGE TWO (20 POWER CELLS)
Also, the game manual (Table 4-2 Point Values) details that a robot (or robots) can also score Ranking Points (RP) by:
- SHIELD GENERATOR OPERATIONAL (1 RP)
- SHIELD GENERATOR ENERGIZED (1 RP)
Treating the robot as a rational agent, it chooses to perform the actions with the optimal expected outcomes and turns and travels to the TRENCH RUN (~7″ at 10ft/s is 1s), then it traverses the TRENCH RUN (~2s), collects the (5) POWER CELLS (5s), then travels back through the TRENCH RUN (~2s), and travels back to the INITIATION LINE (1s). Finally, it shoots the the (5) power cells (5s) in the INNER PORT. This cycle takes 16s and scores us 15 pts (count of 13 cells).
Now that we are located once again at the INITIATION LINE and the POWER CELLS are collected from the TRENCH RUN and SHIELD GENERATOR, we have to travel to our LOAD STATION for POWER CELLS. For this we travel to the TRENCH RUN (1s), through the TRENCH RUN (2s), and then to the LOAD STATION (2s) to collect (5) POWER CELLS (5s). Next we will return through the same path (5s) back to the INITIAION LINE and score the (5) POWER CELLS (5s). This cycle takes 20s (total time 36s, total count ) and scores us 15 pts (count of 18 cells).
With this data, we can extrapolate that one cycle (collecting an item from the LOADING STATION and depositing it at the INNER PORT) takes 20s.
One small note, that in the previous run, we have enough POWER CELLS to hit STAGE ONE and can complete the ROTATIONAL on the CONTROL PANEL We will assume this takes 5s and will score us 10 pts. This takes our total time to 41s.
With a known cycle time of 20s, we can complete another cycle, scoring another 15 pts, and bringing our total time to 61s (count of 23 cells).
While beginning our next cycle, we can stop and complete the POSITIONAL on the CONTROL PANEL. We will assume this takes 5s and will score us 20 pts. This takes total time to 66s.
Now, when we finish this cycle, we score another 15 pts, and our total time is brought to 86s (count of 28 cells).
We can complete two more cycles, scoring 45 pts, which brings our total time to 126s (count of 38 cells).
We now have 9s left as we head into the END GAME.
During the TELEOP period we have scored (30) POWER CELLS and accomplished the ROTATIONAL and POSITIONAL challenges.
Therefore, the MAX TELEOP Mode using this model achieves a score of 120 pts.
We turn and drive to the SHIELD GENERATOR (1s), and hang which we assume will take (5s). This takes 6s (total time 132) and scores us 40 pts.
NOTE:While we assume we are playing “by ourselves” in an imagined scenario where both of the other robots are just KOP Chassis bots, while we can safely assume in any given match that we will still gain at least 5pts each from the other bots driving onto RENDEZVOUS POINT we don’t have enough for the SHIELD GENERATOR OPERATIONAL.
This MAX End Game achieves a score of 40 pts.
- MAX AUTO Mode: 53 pts
- MAX TELEOP Mode: 120 pts
- MAX End Game: 40 pts
MAX TOTAL: 213 pts
So what good has this exercise done? By walking through the match as a rational agent, the flow of a match can be better understood. By picking the optimal actions, necessary robot functions begin to take shape. By approaching the game systematically, rules and strategic advantages provided by those rules can be uncovered.
This exercise has also provided a baseline to evaluate other game strategies against and to help define robot functions from.
Check out Part 2 – Strategy and Research tomorrow for more fun!
If you want to learn more about this process, check out our presentation from the 2018 Purdue FIRST Forums on Robot Requirements!