Why will these steps NOT fix the problems with LIBOR? I am new to finance and recently came across the LIBOR scandal that happened some time in 2008. From my limited understanding of finance, it looks like a scandal that was obviously going to happen and the fix also seems obvious. Clearly, the fix I suggest has issues, or else they would've already been implemented. I want to understand why my proposed solutions will not work.

Here's some background on LIBOR and the scandal. LIBOR is an index maintained by Thomson Reuters on behalf of the British Bankers' Association that attempts to track the interest rate at which world's major banks are borrowing and lending from each other on a daily basis. It is calculated by asking 18 main banks the question "At what rate could you borrow funds, were you to do so by asking for and then accepting interbank offers in a reasonable market size just prior to 11am?" and then taking the average of the numbers reported after removing the top 4 and the bottom 4. Some time in 2008, the people responsible for answering the question started collaborating with each other and reporting incorrect interest rates in order to manipulate Thomson Reuters' calculations. This was good for the participating banks but bad for the world in general since several financial products traded throughout the world are defined in terms of the LIBOR rate and thus misreporting could potentially lead to harmful effects worldwide.

From what I understand after reading many articles, the two main incentives that lead a bank to misreport its borrowing rates are: 1) reporting a higher borrowing rate is bad for your own reputation since if everyone is charging you high rates on loans then it means they don't trust you, 2) if you are one of those 18 banks that are polled for calculating LIBOR and you also happen to be dealing in a financial product that is defined based on LIBOR, then you might be incentivised to manipulate it.

Here is my proposed solution. Following things can be done to handle #1:

Anonymize the answers. Thomson Reuters doesn't need to know exactly which bank reported what. All it needs is the 18 numbers from 18 different banks. Anonymizing should be easy to do with the help of technology. Or law, i.e., Thomson Reuters can promise to not reveal the data to anyone. The technological solution will probably be less prone to future scandals than the legal one.

Instead of asking what rate a bank can borrow at, ask them what rate they will lend at. Of course you will need to specify "to which bank". So do this for each pair of banks. Ask each bank A what rate it will lend at to each bank B. You can't manipulate your own image by answering incorrectly. The only thing you can potentially do is manipulate another bank's image. But with roughly 18^2 numbers to look at, I am sure you can detect a lot of manipulation attempts. For someone to make one specific bank to look bad, they will have to collaborate with several banks at once and ask them to report poor lending rates for that specific bank.

To handle #2, the obvious way is to take the median instead of the mean. Median is much more resilient to manipulation than mean. All 18 banks will have to collaborate to be able to manipulate the median. Is it possible that all of them want LIBOR rate to move in the same direction? Even if it is, I imagine it will be much harder for all of them to collaborate at once.

Finally, one solution to all these problems is to base LIBOR on the actual rates at which trades are being made as opposed to the estimates of those rates. I guess this one will be hard to implement because if the trades are happening over the counter then the records will not be public. And there is probably no way to force the banks to trade on some sort of exchange without invoking law, which as we have seen, is always open to manipulation.
Rather than answer regarding your specific proposal, as that would be of less use generally than providing the current state of thought about interest rate benchmarks, I will provide a quick review of the literature, allowing you to think more generally about the issue. 

Regarding LIBOR specifically, the Wheatley Review of LIBOR was the foundational document outlining the general issues with LIBOR and setting a path for reform. After this review, the International Organization of Securities Commissions (IOSCO) published a report, "Principles for Financial Benchmarks" that addresses general issues surrounding benchmark governance, quality, methodology, and accountability. You will find that this report is particularly important in thinking about benchmark calculation approaches, as it provides a framework for evaluation of benchmark proposals, yours included.

Since then, the Financial Stability Board's Official Sector Steering Group has issued specific recommendations relating to LIBOR reform; this document relates the general IOSCO principles to the specific issues that arise in interest rate benchmarks.

Finally, two academic articles may prove of interest to you. 

Duffie and Dworczak's "Robust Benchmark Design" discusses the design of robust estimators in situations where a benchmark is based on transactions and agents attempt to manipulate these transactions to profit on derivatives based on the benchmark. 

Finally, Duffie and Stein discuss LIBOR reform and alternative reference rates in "Reforming LIBOR and Other Financial-Market Benchmarks."